255
Documenta Math.
On the Classification
of Simple Inductive Limit C ∗ -Algebras, I:
The Reduction Theorem
Dedicated to Professor Ronald G. Douglas
on the occasion of his sixtieth birthday
Guihua Gong1
Received: October 4, 2002
Communicated by Joachim Cuntz
Abstract. Suppose that
A = lim (An =
n→∞
tn
M
M[n,i] (C(Xn,i )), φn,m )
i=1
is a simple C ∗ -algebra, where Xn,i are compact metrizable spaces of
uniformly bounded dimensions (this restriction can be relaxed to a
condition of very slow dimension growth). It is proved in this article
that A can be written as an inductive limit of direct sums of matrix
algebras over certain special 3-dimensional spaces. As a consequence
it is shown that this class of inductive limit C ∗ -algebras is classified
by the Elliott invariant — consisting of the ordered K-group and the
tracial state space — in a subsequent paper joint with G. Elliott and
L. Li (Part II of this series). (Note that the C ∗ -algebras in this class
do not enjoy the real rank zero property.)
1 This material is based upon work supported by, or in part by, the U.S. Army Research
Office under grant number DAAD19-00-1-0152. The research is also partially supported by
NSF grant DMS 9401515, 9622250, 9970840 and 0200739.
Documenta Mathematica 7 (2002) 255–461
256
Guihua Gong
Contents
§0
§1
§2
§3
§4
§5
§6
0
Introduction.
Preparation and some preliminary ideas.
Spectral multiplicity.
Combinatorial results.
Decomposition theorems.
Almost multiplicative maps.
The proof of main theorem.
Introduction
In this article and the subsequent article [EGL], we will classify all the unital
simple C ∗ -algebras A, which can be written as the inductive limit of a sequence
t1
M
i=1
φ1,2
P1,i M[1,i] (C(X1,i ))P1,i −→
t2
M
i=1
φ2,3
P2,i M[2,i] (C(X2,i ))P2,i −→ · · · ,
where Xn,i are compact metrizable spaces with sup{dim Xn,i }n,i < +∞, [n, i]
and tn are positive integers, and Pn,i ∈ M[n,i] (C(Xn,i )) are projections. The
invariant consists of the ordered K-group and the space of traces on the algebra.
The main result in the present article is that a C ∗ -algebra A as above can
be written in another way as an inductive limit so that all the spaces Xn,i
appearing are certain special simplicial complexes of dimension at most three.
Then, in [EGL], the classification theorem will be proved by assuming the C ∗ algebras are such special inductive limits.
In the special case that the groups K∗ (C(Xn,i )) are torsion free, the C ∗ -algebra
A can be written as an inductive limit of direct sums of matrix algebras over
C(S 1 )(i.e., one can replace Xn,i by S 1 ). Combining this result with [Ell2],
without [EGL]–the part II of this series—, we can still obtain the classification
theorem for this special case, which is a generalization of the result of Li for
the case that dim(Xn,i ) = 1 (see [Li1-3]).
The theory of C ∗ -algebras can be regarded as noncommutative topology, and
has broad applications in different areas of mathematics and physics (e.g., the
study of foliated spaces, manifolds with group actions; see [Con]).
One extreme class of C ∗ -algebras is the class of commutative C ∗ -algebras,
which corresponds to the category of ordinary locally compact Hausdorff topological spaces. The other extreme, which is of great importance, is the class of
simple C ∗ -algebras, which must be considered to be highly noncommutative.
For example, the (reduced) foliation C ∗ -algebra of a foliated space is simple if
and only if every leaf is dense in the total space; the cross product C ∗ -algebra,
for a ZZ action on a space X, is simple if and only if the action is minimal.
Even though the commutative C ∗ -algebras and the simple C ∗ -algebras are
opposite extremes, remarkably, many (unital or nonunital) simple C ∗ -algebras
(including the foliation C ∗ -algebra of a Kronecker foliation, see [EE]) have
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
257
been proved to be inductive limits of direct sums of matrix algebras over commutative C ∗L
-algebras, i.e., to be of the form
tn
lim (An = i=1
M[n,i] (C(Xn,i )), φn,m ). (Note that the only commutative C ∗ n→∞
algebras, or matrix algebras over commutative C ∗ -algebras, which are simple
| or Mk (C
| ).) In general, it is a conjecture that any
are the very trivial ones, C
stably finite, simple, separable, amenable C ∗ -algebra is an inductive limit of
subalgebras of matrix algebras over commutative C ∗ -algebras. This conjecture
would be analogous to the result of Connes and Haagerup that any amenable
von Neumann algebra is generated by an upward directed family of sub von
Neumann algebras of type I.
The sweeping classification project of G. Elliott is aimed at the complete classification of simple, separable, amenable C ∗ -algebras in terms of a certain
simple invariant, as we mentioned above, consisting of the ordered K-group
and the space of traces on the
Ltnalgebra. Naturally, the class of inductive limit
C ∗ -algebras A = lim(An = i=1
Pn,i M[n,i] (C(Xn,i ))Pn,i , φn,m ), considered in
→
this article, is an essential ingredient of the project. Following Blackadar [Bl1],
we will call such inductive limit algebras AH algebras.
The study of AH algebras has its roots in the theory of AF algebras (see [Br]
and [Ell4]). But the modern classification theory of AH algebras was inspired
by the seminal paper [Bl3] of B. Blackadar and was initiated by Elliott in [Ell5].
The real rank of a C ∗ -algebra is the noncommutative counterpart of the dimension of a topological space. Until recently, the only known possibilities for
the real rank of a simple C ∗ -algebra were zero or one. It was proved in [DNNP]
that any simple AH algebra
A = lim(An =
→
tn
M
M[n,i] (C(Xn,i )), φn,m )
i=1
has real rank either zero or one, provided that sup{dim Xn,i }n,i < +∞. (Recently, Villadsen has found a simple C ∗ -algebra with real rank different from
zero and one, see [Vi 2].)
For the case of simple C ∗ -algebras of real rank zero, the classification is
quite successful and satisfactory, even though the problem is still not completely solved. Namely, on one hand, the remarkable result of Kirchberg [Kir]
and Phillips [Phi1] completely classified all purely infinite, simple, separable,
amenable C ∗ -algebras with the so called UCT property (see also [R] for an
important earlier result). All purely infinite simple C ∗ -algebras are of real
rank zero; see [Zh]. On the other hand, in [EG1-2] Elliott and the author
∗
completely classified all the stably finite, simple,
Ltn real rank zero C -algebras
which are AH algebras of the form lim(An = i=1 M[n,i] (C(Xn,i )), φn,m ) with
→
dim(Xn,i ) ≤ 3. It was proved by Dadarlat and the author that this class
includes all simple real rank zero AH algebras with arbitrary but uniformly
bounded dimensions for the spaces Xn,i (see [D1-2], [G1-4] and [DG]).
In this article, the AH algebras considered are not assumed to have real rank
zero. As pointed out above, they must have real rank either zero or one. In fact,
Documenta Mathematica 7 (2002) 255–461
258
Guihua Gong
in a strong sense, almost all of them have real rank one. The real rank zero C ∗ algebras are the very special ones for which the space of traces (one part of the
invariant mentioned above) is completely determined by the ordered K-group
of the C ∗ -algebra (the other part of the invariant). Not surprisingly, the lack
of the real rank zero property presents new essential difficulties. Presumably,
dimension one noncommutative spaces are much richer and more complicated
than dimension zero noncommutative spaces. In what follows, we would like
to explain one of the main differences between the real rank zero case and the
general case in the setting of simple AH algebras.
Ltn
Pn,i M[n,i] (C(Xn,i ))Pn,i , φn,m ) is of real rank zero, then
If A = lim(An = i=1
→
Elliott and the author proved a decomposition result (see Theorem 2.21 of
[EG2]) which says that φn,m (for m large enough) can be approximately decomposed as a sum of two parts, φ1 ⊕ φ2 ; one part, φ1 , having a very small
support projection, and the other part, φ2 , factoring through a finite dimensional algebra.
In §4 of the present paper, we will prove a decomposition theorem which
says that, for the simple AH algebra A above (with or without the real rank
zero condition), φn,m (for m large enough) can be approximately decomposed
as a sum of three parts, φ1 ⊕ φ2 ⊕ φ3 : the part φ1 having a very small support
projection compared with the part φ2 ; the part φ2 factoring through a finite
dimensional algebra; and the third part φ3 factoring through a direct sum of
matrix algebras over the interval [0, 1]. (Note that, in the case of a real rank
zero inductive limit, the part φ3 does not appear. In the general case, though,
the part φ3 has a very large support projection compared with the part φ1 ⊕φ2 .)
With this decomposition theorem, we can often deal with the part φ1 ⊕ φ2 by
using the techniques developed in the classification of the real rank zero case
(see [EG1-2], [G1-4], [D1-2] and [DG]).
This new decomposition theorem is much deeper. It reflects the real rank
one (as opposed to real rank zero) property of the simple C ∗ -algebra. The
special case of the decomposition result that the spaces Xn,i are already supposed to be one-dimensional spaces is due to L. Li (see [Li3]). The proof for the
case of higher dimensional spaces is essentially more difficult. In particular, as
preparation, we need to prove certain combinatorial results (see §3) and also
the following result (see §2): Any homomorphism from C(X) to Mk (C(Y ))
can be perturbed to a homomorphism whose maximum spectral multiplicity (for the definition of this terminology, see 1.2.4 below) is not larger than
dim X + dim Y , provided that X 6= {pt} and X is path connected.
The special simplicial complexes used in our main reduction theorem are
the following spaces: {pt}, [0, 1], S 1 , S 2 , {TII,k }∞
and {TIII,k }∞
k=2 ,
k=2 ,
where the spaces TII,k are two-dimensional connected simplicial complexes
with H 1 (TII,k ) = 0 and H 2 (TII,k ) = ZZ/k, and the spaces TIII,k are threedimensional connected simplicial complexes with H 1 (TIII,k ) = 0 = H 2 (TIII,k )
and H 3 (TIII,k ) = ZZ/k. (See 4.2 of [EG2] for details.)
The spaces TII,k and TIII,k are needed to produce the torsion part of
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
259
the K-groups of the inductive limit C ∗ -algebras.
Since the algebras
C(TII,k ), C(TIII,k ), and C(S 2 ) are not stably generated (see [Lo]), difficulties
occur in the construction of homomorphisms from these C ∗ -algebras, when we
prove our main reduction theorem (and the isomorphism theorem in [EGL]).
In the case of real rank zero algebras, this difficulty can be avoided by using
unsuspended E-theory (see [D1-2] and [G1-4]) combined with a certain uniqueness theorem — Theorem 2.29 of [EG2], which only involves homomorphisms
(instead of general completely positive linear maps). Roughly speaking, the
trouble is that a completely positive linear ∗-contraction, which is an “almost
homomorphism”—a G-δ multiplicative map (see 1.1.2 below for the definition
of this concept) for sufficiently large G and sufficiently small δ—, may not be
automatically close to a homomorphism. As we mentioned above, after we approximately decompose φn,m as φ1 ⊕φ2 ⊕φ3 , we will deal with the part φ1 ⊕φ2 ,
by using the results and techniques from the real rank zero case, in particular by using Theorem 1.6.9 below—a strengthened version of Theorem 2.29 of
[EG2]. Therefore, we will consider the composition of the map φ1 ⊕ φ2 and a
homomorphism from a matrix algebra over {pt}, [0, 1], S 1 , S 2 , {TII,k }∞
k=2 ,
and {TIII,k }∞
,
to
A
.
We
need
this
composition
to
be
close
to
a
homomorn
k=2
phism, but φ1 ⊕φ2 is not supposed to be close to a homomorphism (it is close to
the homomorphism φn,m in the case of real rank zero). To overcome the above
difficulty, we prove a theorem in §5—a kind of uniqueness theorem, which may
be roughly described as follows:
For any ε > 0, positive integer N , and finite set F ⊂ A = Mk (C(X)), where
X is one of the spaces {pt}, [0, 1], TII,k , TIII,k , and S 2 , there are a number
δ > 0, a finite set G ⊂ A, and a positive integer L, such that for any two
G-δ multiplicative ( see 1.1.2 below), completely positive, linear ∗-contractions
φ, ψ : Mk (C(X)) → B = Ml (C(Y )) (where dim(Y ) ≤ N ), if they define the
same map on the level of K-theory and also mod-p K-theory (this statement
will be made precise in §5), then there are a homomorphism λ : A → ML (B)
with finite dimensional image and a unitary u ∈ ML+1 (B) such that
k(φ ⊕ λ)(f ) − u(ψ ⊕ λ)(f )u∗ k < ε
for all f ∈ F .
This result is quite nontrivial, and may be expected to have more general applications. Some similar results appear in the literature (e.g., [EGLP, 3.1.4],
[D1, Thm A], [G4, 3.9]). But even for ∗-homomorphisms (which are G-δ multiplicative for any G and δ), all these results (except for contractible spaces)
require that the number L, the size of the matrix, depends on the maps φ and
ψ.
Note that the theorem stated above does not hold if one replaces X by S 1 ,
even if both φ and ψ are ∗-homomorphisms. (Fortunately, we do not need the
theorem for S 1 in this article, since C(S 1 ) is stably generated. But on the
other hand, the lack of such a theorem for S 1 causes a major difficulty in the
formulation and the proof of the uniqueness theorem involving homomorphisms
from C(S 1 ) to Mk (C(X)), in [EGL]— part 2 of this series.)
Documenta Mathematica 7 (2002) 255–461
260
Guihua Gong
With the above theorem, if a G-δ multiplicative, positive, linear ∗-contraction
φ and a ∗-homomorphism (name it ψ) define the same map on the level of Ktheory and mod-p K-theory, then φ⊕λ is close to a ∗-homomorphism (e.g., Adu◦
(ψ ⊕ λ)) for some ∗-homomorphism λ : A → ML (B) with finite dimensional
image. In particular, the size L of the ∗-homomorphism λ can be controlled.
This is essential for the construction of ∗-homomorphisms from A = M k (C(X)),
where X is one of TII,k , TIII,k , and S 2 . In particular, once L is fixed, we can
construct the decomposition of φn,m as φ1 ⊕ φ2 ⊕ φ3 , as mentioned above, such
that the supporting projection of the part φ2 is larger than the supporting
projection of the part φ1 by the amplification of L times. Hence we can prove
that, the composition of the map φ1 ⊕ φ2 and a homomorphism from a matrix
algebra over TII,k , TIII,k , and S 2 to An , is close to a homomorphism (see
Theorems 5.32a and 5.32b below for details).
The theorem is also true for a general finite CW complex X, provided that
K1 (C(X)) is a torsion group.
(Note that for the space S 1 (or the spaces {pt}, [0, 1]), we do not need such
a theorem, since any G-δ multiplicative, positive, linear ∗-contraction from
Mk (C(S 1 )) will automatically be close to a ∗-homomorphism if G is sufficiently
large and δ is sufficiently small.)
The above mentioned theorem and the decomposition theorem both play important roles in the proof of our main reduction theorem, and also in the proof
of the isomorphism theorem in [EGL].
The main results of this article and [EGL] were announced in [G1] and in Elliott’s lecture at the International Congress of Mathematicians in Zurich (see
[Ell3]). Since then, several classes of simple inductive limit C ∗ -algebras have
been classified (see [EGJS], [JS 1-2], and [Th1]). But all these later results
involve only inductive limits of subhomogeneous algebras with 1-dimensional
spectra. In particular, the K0 -groups have to be torsion free, since it is impossible to produce the torsion in K0 -group with one-dimensional spectra alone,
even with subhomogeneous building blocks.
This article is organized as follows. In §1, we will introduce some notations,
collect some known results, prove some preliminary results, and discuss some
important preliminary ideas, which will be used in other sections. In particular,
in §1.5, we will discuss the general strategy in the proof of the decomposition
theorem, of which, the detailed proof will be given in in §2, §3 and §4. In §1.6,
we will prove some uniqueness theorem and factorization theorem which are
important in the proof of the main theorem. Even though the results in §1.6 are
new, most of the methods are modification of known techniques from [EG2],
[D2], [G4] and [DG]. In §2, we will prove the result about maximum spectral
multiplicities, which will be used in §4 and other papers. In §3, we will prove
certain results of a combinatorial nature. In §4, we will combine the results
from §2, §3, and the results in [Li2], to prove the decomposition theorem. In §5,
we will prove the result mentioned above concerning G-δ multiplicative maps.
In §6, we will use §4, §5 and §1.6 to prove our main reduction theorem. Our
main result can be generalized from the case of no dimension growth (i.e., X n,i
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
261
have uniformly bounded dimensions) to the case of very slow dimension growth.
Since the proof of this general case is much more tedious and complicated, we
will deal with this generalization in [G5], which can be regarded as an appendix
to this article.
Acknowledgements. The author would like to thank Professors M. Dadarlat, G. Elliott, L. Li, and H. Lin for helpful conversations. The author also like
to thank G. Elliott, L. Li and H. Lin for reading the article and making suggestions to improve the readability of the article. In particular, L. Li suggested
to the author to make the pictures (e.g., in 3.6, 3.10 and 6.3) to explain the
ideas in the proof of some results; G. Elliott suggested to the author to write
a subsection §1.5 to explain the general strategy for proving a decomposition
theorem.
1
Preparation and some preliminary ideas
We will introduce some conventions, general assumptions, and preliminary results in this section.
1.1
General assumptions on inductive limits
1.1.1. If A and B are two C ∗ -algebras, we use Map(A, B) to denote the
space of all linear, completely positive ∗-contractions from A to
B. If both A and B are unital, then Map(A, B)1 will denote the subset of
Map(A, B) consisting of unital maps. By word “map”, we shall mean linear,
completely positive ∗-contraction between C ∗ -algebras, or else we shall mean
continuous map between topological spaces, which one will be clear from the
context.
By a homomorphism between C ∗ -algebras, will be meant a ∗homomorphism. Let Hom(A, B) denote the space of all homomorphisms
from A to B. Similarly, if both A and B are unital, let Hom(A, B)1 denote the
subset of Hom(A, B) consisting of unital homomorphisms.
Definition 1.1.2. Let G ⊂ A be a finite set and δ > 0. We shall say that
φ ∈ Map(A, B) is G-δ multiplicative if
kφ(ab) − φ(a)φ(b)k < δ
for all a, b ∈ G.
Sometimes, we use MapG−δ (A, B) to denote all the G-δ multiplicative maps.
1.1.3. In the notation for an inductive system (An , φn,m ), we understand
that φn,m = φm−1,m ◦ φm−2,m−1 · · · ◦ φn,n+1 , where all φn,m : An → Am are
homomorphisms.
Documenta Mathematica 7 (2002) 255–461
262
Guihua Gong
Ltn
Ain ,
We shall assume that, for any summand Ain in the direct sum An = i=1
i
necessarily, φn,n+1 (1Ain ) 6= 0, since, otherwise, we could simply delete An from
An without changing the limit algebra.
L
L
1.1.4. If An = i Ain and Am = j Ajm , we use φi,j
n,m to denote the partial
i
map of φn,m from the i-th block An of An to the j-th block Ajm of Am .
In this article, we will assume that all inductive limit C ∗ -algebras are simple.
That is, the limit algebra has no nontrivial proper closed two sided ideals.
We will also assume that every inductive limit C ∗ -algebra A = lim(An , φn,m )
→
| ) (the matrix algebra
coming into consideration is different both from Mk (C
|
over C), and from K(H)L(the algebra of all compact operators).
i
Since A = lim(An =
i An , φn,m ) is simple, by 5.3.2(b) of [DN], we may
→
i
j
assume that φi,j
n,m (1Ain ) 6= 0 for any blocks An and Am , where n < m.
1.1.5.
To avoid certain counter examples (see [V]) of the main result of
this article, we will restrict our attention, in this article, to inductive systems
satisfying the following very slow dimension growth condition. This is
a strengthened form of the condition of slow dimension growth introduced in
[BDR].
Ltn
Pn,i M[n,i] (C(Xn,i ))Pn,i , φn,m ) is a unital inductive limit sysIf lim(An = i=1
→
tem, the very slow dimension growth condition is
¾
½
(dim Xn,i )3
= 0,
lim max
n→+∞
i
rank(Pn,i )
where dim(Xn,i ) denotes the (covering) dimension of Xn,i .
In this article, we will also study non-unital inductive limit algebras. The
above formula must then be slightly modified. The very slow dimension growth
condition in the non-unital case is that, for any summand
Ain = Pn,i M[n,i] (C(Xn,i ))Pn,i of a fixed An ,
lim
m→+∞
max
i,j
(
(dim Xm,j )3
rankφi,j
n,m (1Ain )
)
= 0,
i
j
where φi,j
n,m is the partial map of φn,m from An to Am .
(For a unital inductive limit, the two conditions above are equivalent. Of
course, both conditions are only proposed for the simple case.)
If the set {dim Xn,i } is bounded, i.e, there is an M such that
dim Xn,i ≤ M
for all n and i, then the inductive system automatically satisfies the very slow
dimension growth condition, as we already assume that the limit algebra is not
| ) or K(H).
Mk (C
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
263
We will prove our main reduction theorem for the case of uniformly bounded
dimensions in this article, since it is significantly simpler than the case of very
slow dimension growth. The general case will be discussed in [G5]—an appendix of this article. But the decomposition theorem will be proved for the
case of very slow dimension growth.
It must be noted that, without the above assumption on dimension growth,
the main theorem of this article does not hold (see [Vi1]). We shall leave the
following question open: can the above condition of very slow dimension growth
be replaced by the similar (but weaker) condition of slow dimension growth (see
[BDR]), in the main theorem of this article?
1.1.6. By 2.3 of [Bl1], in the inductive limit
A = lim(An =
→
tn
M
Pn,i M[n,i] (C(Xn,i ))Pn,i , φn,m ),
i=1
one can always replace the compact metrizable spaces Xn,i by finite simplicial
complexes. Note that the replacement does not increase the dimensions of
the spaces. Therefore, in this article, we will always assume that all
the spaces Xn,i in a given inductive system are finite simplicial
complexes. Also, we will further assume that all Xn,i are path
connected. Otherwise, we will separate different components into different
direct summands. (Note that a finite simplicial complex has at most finitely
many path connected components.)
By simplicial complex we mean finite simplicial complex or polyhedron; see
[St].
1.1.7.
(a) We use the notation #(·) to denote the cardinal number of the set, if the
argument is a finite set. Very often, the sets under consideration will be sets
with multiplicity, and then we shall also count multiplicity when we use the
notation #.
(b) We shall use a∼k to denote a, · · · , a . For example,
| {z }
k copies
{a∼3 , b∼2 } = {a, a, a, b, b}.
(c) int(·) is used to denote the integer part of a real number. We reserve the
notation [·] for equivalence classes in possibly different contexts.
(d) For any metric space X, any x0 ∈ X and any c > 0, let Bc (x0 ) := {x ∈
X | d(x, x0 ) < c} denote the open ball with radius c and centre x0 .
(e) Suppose that A is a C ∗ -algebra, B ⊂ A is a subalgebra, F ⊂ A is a (finite)
subset and let ε > 0. If for each element f ∈ F , there is an element g ∈ B such
that kf − gk < ε, then we shall say that F is approximately contained in B to
within ε, and denote this by F ⊂ε B.
Documenta Mathematica 7 (2002) 255–461
264
Guihua Gong
(f) Let X be a compact metric space. For any δ > 0, a finite set {x1 , x2 , · · · xn }
is said to be δ-dense in X, if for any x ∈ X, there is xi such that dist(x, xi ) < δ.
(g) We shall use • to denote any possible positive integer. To save notation,
y, y ′ , y ′′ , · · · or a1 , a2 , · · · may be used for finite sequences if we do not care how
many terms are in the sequence. Similarly, A1 ∪ A2 ∪ · · · or A1 ∩ A2 ∩ · · · may be
used for finite union or finite intersection. If there is a danger of confusion with
infinite sequence, union, or intersection, we will write them as a1 , a2 , · · · , a• ,
A1 ∪ A2 ∪ · · ·L
∪ A• , or A1 ∩ A2 ∩ · · · A• .
t
(h) For A = i=1 Mki (C(Xi )), where Xi are path connected simplicial comLt
| ), which could be conplexes, we use the notation r(A) to denote i=1 Mki (C
sidered to be a subalgebra of A consisting of t-tuples of constant functions from
| ) (i = 1, 2, · · · , t). Fix a base point x0 ∈ Xi for each Xi , one can
Xi to Mki (C
i
define a map r : A → r(A) by
r(f1 , f2 , · · · , ft ) = (f1 (x01 ), f2 (x02 ), · · · ft (x0t )) ∈ r(A).
(i) For any two projections p, q ∈ A, we use the notation [p] ≤ [q] to denote
that p is unitarily equivalent to a sub projection of q. And we use p ∼ q to
denote that p is unitarily equivalent to q.
1.2
Spectrum and spectral variation of a homomorphism
1.2.1.
Let Y be a compact metrizable space. Let P ∈ Mk1 (C(Y )) be a
| )
projection with rank(P ) = k ≤ k1 . For each y, there is a unitary uy ∈ Mk1 (C
(depending on y) such that
P (y) = uy
1
..
.
1
0
..
.
0
∗
uy ,
where there are k 1’s on the diagonal. If the unitary uy can be chosen to be
continuous in y, then P is called a trivial projection.
It is well known that any projection P ∈ Mk1 (C(Y )) is locally trivial. That
is, for any y0 ∈ Y , there is an open set Uy0 ∋ y0 , and there is a continuous
unitary-valued function
| )
u : Uy0 → Mk1 (C
such that the above equation holds for u(y) (in place of uy ) for any y ∈ Uy0 .
If P is trivial, then P Mk1 (C(X))P ∼
= Mk (C(X)).
1.2.2. Let X be a compact metrizable space and ψ : C(X) → P Mk1 (C(Y ))P
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
265
be a unital homomorphism. For any given point y ∈ Y , there are points
| ) such that
x1 (y), x2 (y), · · · , xk (y) ∈ X, and a unitary Uy ∈ Mk1 (C
f (x1 (y))
..
ψ(f )(y) = P (y)Uy
.
f (xk (y))
0
..
.
∗
Uy P (y) ∈ P (y)Mk1 (C
| )P (y)
0
for all f ∈ C(X). Equivalently, there are k rank one orthogonal projections
Pk
p1 , p2 , · · · , pk with i=1 pi = P (y) and x1 (y), x2 (y), · · · , xk (y) ∈ X, such that
ψ(f )(y) =
k
X
f (xi (y))pi ,
i=1
∀ f ∈ C(X).
Let us denote the set {x1 (y), x2 (y), · · · , xk (y)}, counting multiplicities, by
SPψy . In other words, if a point is repeated in the diagonal of the above
matrix, it is included with the same multiplicity in SPψy . We shall call
SPψy the spectrum of ψ at the point y. Let us define the spectrum of
ψ, denoted by SPψ, to be the closed subset
SPψ :=
[
y∈Y
SPψy ⊆ X.
Alternatively, SPψ is the complement of the spectrum of the kernel of ψ, considered as a closed ideal of C(X). The map ψ can be factored as
i∗
ψ1
C(X) −→ C(SPψ) −→ P Mk1 (C(Y ))P
with ψ1 an injective homomorphism, where i denotes the inclusion SPψ ֒→ X.
Also, if A = P Mk1 (C(Y ))P , then we shall call the space Y the spectrum of
the algebra A, and write SPA = Y (= SP(id)).
1.2.3. In 1.2.2, if we group together all the repeated points in
{x1 (y), x2 (y), · · · , xk (y)}, and sum their corresponding projections, we can
write
l
X
f (λi (y))Pi
(l ≤ k),
ψ(f )(y) =
i=1
where {λ1 (y), λ2 (y), · · · , λl (y)} is equal to {x1 (y), x2 (y), · · · , xk (y)} as a set,
but λi (y) 6= λj (y) if i 6= j; and each Pi is the sum of the projections corresponding to λi (y). If λi (y) has multiplicity m (i.e., it appears m times in
{x1 (y), x2 (y), · · · , xk (y)}), then rank(Pi ) = m.
Documenta Mathematica 7 (2002) 255–461
266
Guihua Gong
Definition 1.2.4. Let ψ, y, and Pi be as above. The maximum spectral
multiplicity of ψ at the point y is defined to be maxi (rankPi ). The
maximum spectral multiplicity of ψ is defined to be the supremum of
the maximum spectral multiplicities of ψ at the various points of Y .
The following result is the main theorem in §2, which says that we can make
the homomorphism not to have too large spectral multiplicities, up to a small
perturbation.
Theorem 2.1. Let X and Y be connected simplicial complexes and X 6= {pt}.
Let l = dim(X) + dim(Y ). For any given finite set G ⊂ C(X), any ε > 0, and
any unital homomorphism φ : C(X) → P M• (C(Y ))P , where P ∈ M• (C(Y ))
is a projection, there is a unital homomorphism φ′ : C(X) → P M• (C(Y ))P
such that
(1) kφ(g) − φ′ (g)k < ε for all g ∈ G;
(2) φ′ has maximum spectral multiplicity at most l.
1.2.5.
Set P k (X) = X × X × · · · × X / ∼, where the equivalence relation ∼
{z
}
|
k
is defined by (x1 , x2 , · · · , xk ) ∼ (x′1 , x′2 , · · · , x′k ) if there is a permutation σ of
{1, 2, · · · , k} such that xi = x′σ(i) , for each 1 ≤ i ≤ k. A metric d on X can be
extended to a metric on P k (X) by
d([x1 , x2 , · · · , xk ], [x′1 , x′2 , · · · , x′k ]) = min max d(xi , x′σ(i) ),
σ
1≤i≤k
where σ is taken from the set of all permutations, and [x1 , · · · , xk ] denotes the
equivalence class in P k (X) of (x1 , · · · , xk ).
1.2.6. Let X be a metric space with metric d. Two k-tuples of (possibly
repeating) points {x1 , x2 , · · · , xk } ⊂ X and {x′1 , x′2 , · · · , x′k } ⊂ X are said to be
paired within η if there is a permutation σ such that
d(xi , x′σ(i) ) < η,
i = 1, 2, · · · , k.
This is equivalent to the following. If one regards (x1 , x2 , · · · , xk ) and
(x′1 , x′2 , · · · , x′k ) as two points in P k X, then
d([x1 , x2 , · · · , xk ], [x′1 , x′2 , · · · , x′k ]) < η.
1.2.7. Let ψ : C(X) → P Mk1 (C(Y ))P be a unital homomorphism as in
1.2.5. Then
ψ ∗ : y 7→ SPψy
defines a map Y → P k X, if one regards SPψy as an element of P k X. This
map is continuous. In term of this map and the metric d, let us define the
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
267
spectral variation of ψ:
SPV(ψ) := diameter of the image of ψ ∗ .
Definition 1.2.8. We shall call the projection Pi in 1.2.3 the spectral
projection of φ at y with respect to the spectral element λi (y). If
X1 ⊂ X is a subset of X, we shall call
X
Pi
λi (y)∈X1
the spectral projection of φ at y corresponding to the subset X1
(or with respect to the subset X1 ).
In general, for an open set U ⊂ X, the spectral projection P (y) of φ at y
corresponding to U does not depend on y continuously. But the following
lemma holds.
Lemma 1.2.9. Let U ⊂ X be an open subset. Let φ : C(X) → M• (C(Y )) be a
homomorphism. Suppose that W ⊂ Y is an open subset such that
SPφy ∩ (U \U ) = ∅, ∀y ∈ W.
Then the function
y 7→ spectral projection of φ at y correponding to U
is a continuous function on W . Furthermore, if W is connected then #(SPφ y ∩
U ) (counting multiplicity) is the same for any y ∈ W , and the map y 7→ SPφy ∩
U ∈ P l X is a continuous map on W , where l = #(SPφy ∩ U ).
Proof: Let P (y) denote the spectral projection of φ at y corresponding to the
open set U . Fix y0 ∈ W . Since SPφy0 is a finite set, there is an open set
U1 ⊂ U1 ⊂ U (⊂ X) such that SPφy0 ∩ U1 = SPφy0 ∩ U (= SPφy0 ∩ U ), or in
other words, SPφy0 ⊂ U1 ∪ (X\U ). Considering the open set U1 ∪ (X\U ), by
the continuity of the function
y 7→ SPφy ∈ P k X,
where k = rank(φ(1)), there is an open set W1 ∋ y0 such that
(1)
SPφy ⊂ U1 ∪ (X\U ),
Let χ ∈ C(X) be a function satisfying
½
1
χ(x) =
0
if
if
∀y ∈ W1 .
x ∈ U1
x ∈ X\U .
Documenta Mathematica 7 (2002) 255–461
268
Guihua Gong
Then from (1) and the definition of spectral projection it follows that
φ(χ)(y) = P (y), ∀y ∈ W1 .
In particular, P (y) is continuous at y0 .
The additional part of the lemma follows from the continuity of P (y) and the
connectedness of W .
⊔
⊓
In the above proof, we used the following fact, a consequence of the continuity
of the map y 7→ SPφy . We state it separately for our future use.
Lemma 1.2.10. Let X be a finite simplicial complex, X1 ⊂ X be a closed
subset, and φ : C(X) → M• (C(Y )) be a homomorphism. For any y0 ∈ Y , if
SPφy0 ∩ X1 = ∅, then there is an open set W ∋ y0 such that SPφy ∩ X1 = ∅
for any y ∈ W .
Another equivalent statement is the following. Let U ⊂ X be an open subset.
For any y0 ∈ Y , if SPφy0 ⊂ U , then there is an open set W ∋ y0 such that
SPφy ⊂ U for any y ∈ W .
1.2.11. In fact the above lemma is a consequence of the following more general
| ) is a homomorphism satisfying SPφ ⊂ U for a
principle: If φ : C(X) → M• (C
| ) which is
certain open set U , then for any homomorphism ψ : C(X) → M• (C
close enough to φ, we have SPψ ⊂ U . We state it as the following lemma.
Lemma 1.2.12. Let F ⊂ C(X) be a finite set of elements which generate
C(X) as a C ∗ -algebra. For any ε > 0, there is a δ > 0 such that if two
| ) satisfy
homomorphisms φ, ψ : C(X) → M• (C
kφ(f ) − ψ(f )k < δ,
∀f ∈ F,
then SPψ and SPφ can be paired within ε. In particular, SPψ ⊂ U , where U
is the open set defined by U = {x ∈ X | ∃x′ ∈ SPφ with dist(x, x′ ) < ε}.
1.2.13. For any C ∗ algebra A (usually we let A = C(X) or A =
P Mk (C(X))P ), any homomorphism φ : A → M• (C(Y )), and any closed subset
Y1 ⊂ Y , denote by φ|Y1 the following composition:
φ
A −→ M• (C(Y ))
restriction
−→
M• (C(Y1 )).
(As usual, for a subset or subalgebra A1 ⊂ A, φ|A1 will be used to denote the
restriction of φ to A1 . We believe that there will be no danger of confusion as
the meaning will be clear from the context.)
The following trivial fact will be used frequently.
Lemma 1.2.14. Let Y1 , Y2 ⊂ Y be two closed subsets. If φ1 : A → Mk (C(Y1 ))
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
269
and φ2 : A → Mk (C(Y2 )) are two homomorphisms with φ1 |Y1 ∩Y2 = φ2 |Y1 ∩Y2 ,
then for any a ∈ A, the matrix-valued function y 7→ φ(a)(y), where
½
φ1 (a)(y)
if y ∈ Y1
φ(a)(y) =
φ2 (a)(y)
if y ∈ Y2 ,
is a continuous function on Y1 ∪ Y2 (i.e., it is an element of Mk (C(Y1 ∪ Y2 )).
Furthermore, a 7→ φ(a) defines a homomorphism φ : A → Mk (C(Y1 ∪ Y2 )).
1.2.15. Let X be a compact connected space and let Q be a projection of rank
n in MN (C(X)). The weak variation of a finite set F ⊂ QMN (C(X))Q
is defined by
ω(F ) = sup
inf
max kuΠ1 (a)u∗ − Π2 (a)k
Π1 ,Π2 u∈U (n) a∈F
where Π1 , Π2 run through the set of irreducible representations of
| ).
QMN (C(X))Q into Mn (C
Let Xi be compact connected
spaces and Qi ∈ Mni (C(Xi )) be projections.
L
For a finite set F ⊂ i Qi Mni (C(X
L i ))Qi , define the weak variation ω(F )
to be maxi ω(πi (F )), where πi : i Qi Mni (C(Xi ))Qi → Qi Mni (C(Xi ))Qi is
the natural project map onto the i-th block.
The set F is said to be weakly approximately constant to within ε if
ω(F ) < ε. The other description of this concept can be found in [EG2, 1.4.11]
(see also [D2, 1.3]).
1.2.16. Let φ : Mk (C(X)) → P Ml (C(Y ))P be a unital homomorphism. Set
φ(e11 ) = p, where e11 is the canonical matrix unit corresponding to the upper
left corner. Set
φ1 = φ|e11 Mk (C(X))e11 : C(X) −→ pMl (C(Y ))p.
Then P Ml (C(Y ))P can be identified with pMl (C(Y ))p ⊗ Mk in such a way
that
φ = φ 1 ⊗ 1k .
Let us define
SPφy := SP(φ1 )y ,
SPφ := SPφ1 ,
SPV(φ) := SPV(φ1 ) .
Suppose that X and Y are connected. Let Q be a projection in Mk (C(X))
and φ : QMk (C(X))Q → P Ml (C(Y ))P be a unital map. By the Dilation
Lemma (2.13 of [EG2]; see Lemma 1.3.1 below), there are an n, a projection
P1 ∈ Mn (C(Y )), and a unital homomorphism
φ̃ : Mk (C(X)) −→ P1 Mn (C(Y ))P1
Documenta Mathematica 7 (2002) 255–461
270
Guihua Gong
such that
φ = φ̃|QMk (C(X))Q .
(Note that this implies that P is a subprojection of P1 .) We define:
SPφy := SPφ̃y ,
SPφ := SPφ̃,
SPV(φ) := SPV(φ̃) .
(Note that these definitions do not depend on the choice of the dilation φ̃.)
The following lemma was essentially proved in [EG2, 3.27] (the additional part
is [EG 1.4.13]).
Lemma 1.2.17. Let X be a path connected compact metric space. Let
p0 , p1 , p2 , · · · , pn ∈ M• (C(Y )) be mutually orthogonal projections such that
rank(pi ) ≥ rank(p0 ), i = 1, 2, · · · , n. Let {x1 , x2 , · · · , xn } be a 2δ -dense subset of X. If a homomorphism φ : C(X) → M• (C(Y )) is defined by
φ(f ) = φ0 (f ) ⊕
n
X
f (xi )pi ,
i=1
where φ0 : C(X) → p0 M• (C(Y ))p0 is an arbitrary homomorphism, then
SPV(φ) < δ. Consequently, if a finite set F ⊂ C(X) satisfies the condition
that kf (x) − f (x′ )k < ε, for any f ∈ F , whenever dist(x, x′ ) < δ, then φ(F ) is
weakly approximately constant to within ε.
(For convenience, we
Pnwill call such a homomorphism ψ : C(X) → M• (C(Y )),
defined by ψ(f ) = i=1 f (xi )pi , a homomorphism defined by point evaluations on the set {x1 , x2 , · · · , xn }.)
Proof: For any two points y, y ′ ∈ Y , the sets SPφy and SPφy′ have the following
subset in common:
∼rank(p1 )
{x1
∼rank(p2 )
, x2
, · · · , xn∼rank(pn ) }.
The remaining parts of SPφy and SPφy′ are SP(φ0 )y and SP(φ0 )y′ , respectively,
which have at most rank(p0 ) elements.
It is easy to prove the following fact.
For any a, b ∈ X, the sets
{a, x1 , x2 , · · · , xn } and {b, x1 , x2 , · · · , xn } can be paired within δ. In fact, by
path connectedness of X and 2δ -density of the set {x1 , x2 , · · · , xn }, one can find
a sequence
a, xj1 , xj2 , · · · , xjk , b
beginning with a and ending with b such that each pair of consecutive terms
has distance smaller than δ. So {a, xj1 , · · · , xjk−1 , xjk } can be paired with
{xj1 , xj2 , · · · , xjk , b} (= {b, xj1 , · · · , xjk }) one by one to within δ. The other
parts of the sets are identical, each element can be paired with itself.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
271
Combining the above fact with the condition that rank(pi ) ≥ rank(p0 ) for any
i, we know that SPφy and SPφy′ can be paired within δ. That is, SPV(φ) < δ.
The rest of the lemma is obvious. Namely, for any two points y, y ′ , φ(f )(y) is
approximately unitarily equivalent to φ(f )(y ′ ) to within ε, by the same unitary
for all f ∈ F (see [EG2, 1.4.13]).
⊔
⊓
1.2.18. In the last part of the above lemma, one does not need φ0 to be
a homomorphism to guarantee φ(F ) to be weakly approximately constant to
within a small number. In fact, the following is true.
Suppose that all the notations are as in 1.2.17 except that the maps φ0 :
C(X) → p0 M• (C(Y ))p0 and φ : C(X) → M• (C(Y )) are no longer homomorphisms. Suppose that for any y ∈ Y , there is a homomorphism
| )p0 (y) such that
ψy : C(X) → p0 (y)M• (C
kφ0 (f )(y) − ψy (f )k < ε, ∀f ∈ F.
Then the set φ(F ) is weakly approximately constant to within 3ε. One can
prove this claim as follows.
| ) by φy (f ) = ψy (f ) ⊕
For any y ∈ Y , define a homomorphism φy → M• (C
P
n
′
f
(x
)p
.
Then
for
any
two
points
y,
y
∈
Y
,
as
same
as in Lemma 1.2.17,
i
i
i=1
SP(φy ) and SP(φy′ ) can be paired within δ. Therefore, φy (f ) is approximately
unitarily equivalent to φy′ (f ) to within ε, by the same unitary for all f ∈ F .
On the other hand,
kφ(f )(y) − φy (f )k < ε and kφ(f )(y ′ ) − φy′ (f )k < ε,
∀f ∈ F.
Hence φ(f )(y) is approximately unitarily equivalent to φ(f )(y ′ ) to within 3ε,
by the same unitary for all f ∈ F .
1.2.19. Suppose that F ⊂ Mk (C(X)) is a finite set and ε > 0. Let F ′ ⊂ C(X)
be the finite set consisting of all entries of elements in F and ε′ = kε , where k
is the order of the matrix algebra Mk (C(X)).
It is well known that, for any k × k matrix a = (aij ) ∈ Mk (B) with entries
aij ∈ B, kak ≤ k maxij kaij k. This implies the following two facts.
Fact 1. If φ1 , ψ1 ∈ Map(C(X), B) are (complete positive) linear ∗-contraction
(as the notation in 1.1.1) which satisfy
kφ1 (f ) − ψ1 (f )k < ε′ , ∀f ∈ F ′ ,
then φ := φ1 ⊗ idk ∈ Map(Mk (C(X)), Mk (B)) and ψ := ψ1 ⊗ idk ∈
Map(Mk (C(X)), Mk (B)) satisfy
kφ(f ) − ψ(f )k < ε, ∀f ∈ F.
Fact 2. Suppose that φ1 ∈ Map(C(X), M• (C(Y ))) is a (complete positive)
linear ∗-contraction. If φ1 (F ′ ) is weakly approximately constant to within ε′ ,
then φ1 ⊗ idk (F ) is weakly approximately constant to within ε.
Documenta Mathematica 7 (2002) 255–461
272
Guihua Gong
Suppose that a homomorphism φ1 ∈ Hom(C(X), B) has a decomposition described as follows. There exist mutually orthogonal projections p1 , p2 ∈ B with
p1 + p2 = 1B and ψ1 ∈ Hom(C(X), p2 Bp2 ) such that
kφ1 (f ) − p1 φ1 (f )p1 ⊕ ψ1 (f )k < ε′ , ∀f ∈ F ′ .
Then there is a decomposition for φ := φ1 ⊗ idk :
kφ(f ) − P1 φ1 (f )P1 ⊕ ψ(f )k < ε, ∀f ∈ F,
where ψ := ψ1 ⊗ idk and P1 = p1 ⊗ 1k .
In particular, if B = M• (C(Y )) and ψ1 is described by
X
ψ1 (f )(y) =
f (αi (y))qi (y), ∀f ∈ C(X),
where
by
P
qi = p2 and αi : Y → X are continuous maps, then ψ can be described
ψ(f )(y) =
X
qi (y) ⊗ f (αi (y)), ∀f ∈ Mk (C(X)),
regarding Mk (M• (C(Y ))) as M• (C(Y )) ⊗ Mk .
If αi are constant maps, the homomorphism ψ1 is called a
defined by point evaluations as in Lemma 1.2.17. In this case,
the above ψ a homomorphism defined by point evaluations.
From the above, we know that to decompose a
φ
∈
Hom(Mk (C(X)), M• (C(Y ))), one only needs
φ1 := φ|e11 Mk (C(X))e11 ∈ Hom(C(X), φ(e11 )M• (C(Y ))φ(e11 )).
1.3
homomorphism
we will also call
homomorphism
to decompose
Full matrix algebras, corners, and the dilation lemma
Some results in this article deal with a corner QMN (C(X))Q of the matrix
algebra MN (C(X)). But using the following lemma and some other techniques,
we can reduce the problems to the case of a full matrix algebra MN (C(X)).
The following dilation lemma is Lemma 2.13 of [EG2].
Lemma 1.3.1. (cf. Lemma 2.13 of [EG2]) Let X and Y be any connected
finite CW complexes. If φ : QMk (C(X))Q → P Mn (C(Y ))P is a unital
homomorphism, then there are an n1 , a projection P1 ∈ Mn1 (C(Y )), and a
unital homomorphism φ̃ : Mk (C(X)) → P1 Mn1 (C(Y ))P1 with the property
that QMk (C(X))Q and P Mn (C(Y ))P can be identified as corner subalgebras
of Mk (C(X)) and P1 Mn1 (C(Y ))P1 respectively (i.e., Q and P can be considered to be subprojections of 1k and P1 , respectively) and, furthermore, in such
a way that φ is the restriction of φ̃.
If φt : QMk (C(X))Q → P Mn (C(Y ))P, (0 ≤ t ≤ 1) is a path of unital
homomorphisms, then there are P1 Mn1 (C(Y ))P1 (as above) and a path of unital
homomorphisms φ̃t : Mk (C(X)) → P1 Mn1 (C(Y ))P1 such that QMk (C(X))Q
and P Mn (C(Y ))P are corner subalgebras of Mk (C(X)) and P1 Mn1 (C(Y ))P1
respectively and φt is the restriction of φ̃t .
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
273
Definition 1.3.2. Let A be a C ∗ -algebra. A sub-C ∗ -algebra A1 ⊂ A will be
called a limit corner subalgebra of A, if there is a sequence of increasing
projections
P1 ≤ P 2 ≤ · · · ≤ P n ≤ · · · ,
S∞
such that A1 = n=1 Pn APn .
Using Lemma 1.3.1, it is routine to prove the following lemma.
Lemma 1.3.3. (cf. 4.24 of [EG2]) For any AH algebra A = lim(An =
→
Ltn
P
M
(C(X
))P
,
φ
),
there
is
an
inductive
limit
Ã
=
lim(Ãn =
n,i
n,i
n,i
n,m
[n,i]
i=1
→
Ltn
M
(C(X
)),
φ̃
)
of
full
matrix
algebras
over
{X
},
such
that
n,i
n,m
n,i
{n,i}
i=1
A is isomorphic to a limit corner subalgebra of Ã. In particular, each
Pn,i M[n,i] (C(Xn,i ))Pn,i is a corner subalgebra of M{n,i} (C(Xn,i )) and φn,m
Ltn
Pn,i M[n,i] (C(Xn,i ))Pn,i .
is the restriction of φ̃n,m on An = i=1
Furthermore, A is stably isomorphic to Ã.
Remark 1.3.4. In the above lemma, in general, the homomorphisms φ̃n,m
cannot be chosen to be unital, even if all the homomorphisms φn,m are unital.
If A is unital, then A can be chosen to be the cut-down of à by a single
projection (rather than a sequence of projections).
Remark 1.3.5. In Lemma 1.3.3, if A is simple and satisfies the very slow dimension growth condition, then so is Ã. Hence when we consider the nonunital
case, we may always assume that A is an inductive limit of direct sums of full
matrix algebras over C(Xn,i ) without loss of generality. Since the reduction
theorem in this paper will be proved without the assumption of unitality, we
may assume that the C ∗ -algebra A is an inductive limit of full matrix algebras
over finite simplicial complexes. But even in this case, we still need to consider
the cut-down P Ml (C(X))P of Ml (C(X)) in some situations, since the image
of a trivial projection may not be trivial.
In the proof of the decomposition theorem in §4, we will not assume that A is
unital but we will assume that A is the inductive limit of full matrix algebras.
Note that a projection in M• (C(X)) corresponds to a complex vector bundle
over X. The following result is well known (see Chapter 8 of [Hu]). This result
is often useful when we reduce the proof of a result involving the cut-down
P Ml (C(X))P to the special case of the full matrix algebra Ml (C(X)).
Lemma 1.3.6. Let X be a connected simplicial complex and P ∈ Ml (C(X)) be
a non-zero projection. Let n = rank(P ) + dim(X) and m = 2 dim(X) + 1.
Then P is Murray-von Neumann equivalent to a subprojection of 1n , and 1n is
Murray-von Neumann equivalent to a subprojection of P ⊕ P ⊕ · · · P , where 1n
{z
}
|
m
is a trivial projection with rank n. Therefore, P Ml (C(X))P can be identified
Documenta Mathematica 7 (2002) 255–461
274
Guihua Gong
as a corner subalgebra of Mn (C(X)), and Mn (C(X)) can be identified as a
corner subalgebra of Mm (P Ml (C(X))P ).
1.4
Topological preliminaries
In this subsection, we will introduce some notations and results in the topology
of simplicial complexes. We will also introduce a well known method for the
construction of cross sections of a fibre bundle. The content of this subsection
may be found in [St], [Hu] and [Wh].
1.4.1. Let X be a connected simplicial complex. Endow X with a metric d as
follows.
For each n-simplex ∆, one can identify ∆ with an n-simplex in IRn whose
edges are of length 1 (of course the identification should preserve the affine
structure of the simplices). (Such a simplex is the convex hull of n + 1 points
{x0 , x1 , · · · , xn } in IRn with dist(xi , xj ) = 1 for any i 6= j ∈ {0, 1, · · · n}.) Such
an identification gives rise to a unique metric on ∆. The restriction of metric
d of X to ∆ is defined to be the above metric for any simplex ∆ ⊂ X. For
any two points x, y ∈ X, d(x, y) is defined to be the length of the shortest
path connecting x and y. (The length is measured in individual simplexes, by
breaking the path into small pieces.)
If X is not connected, denote by L the maximum of the diameters of all the
connected components. Define d(x, y) = L + 1, if x and y are in different components. (Recall that all the simplicial complexes in this article are supposed
to be finite.)
1.4.2. For a simplex ∆, by ∂∆, we denote the boundary of the simplex ∆,
which is the union of all proper faces of ∆. Note that if ∆ is a single point—
zero dimensional simplex, then ∂∆ = ∅. Obviously, dim(∂∆) = dim(∆) − 1.
(We use the standard convention that the dimension of the empty space is −1.)
By interior(∆), we denote ∆\∂∆. Let X be a simplicial complex. Obviously,
for each x ∈ X, there is a unique simplex ∆ such that x ∈ interior(∆), which
is the simplex ∆ of lowest dimension with the condition that x ∈ ∆. (Here
we use the fact that if two different simplices of the same dimension intersect,
then the intersection is a simplex of lower dimension.)
For any simplex ∆, define
[
Star(∆) = {interior(∆′ ) | ∆′ ∩ ∆ 6= ∅}.
Then Star(∆) is an open set which covers ∆.
We will use the following two open covers of the simplicial complex X.
(a) For any vertex x ∈ X, let
[
Wx = Star({x})(= {interior(∆) | x ∈ ∆}).
Obviously {Wx }x∈Vertex(X) is an open cover parameterized by vertices of X.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
275
In this open cover, the intersection Wx1 ∩ Wx2 ∩ · · · ∩ Wxk is nonempty if and
only if x1 , x2 , · · · , xk span a simplex of X.
(b) We denote the original simplicial structure of X by σ. Introduce a barycentric subdivision (X, τ ) of (X, σ).
Then for each simplex ∆ of (X, σ) (before subdivision), there is exactly one
point C∆ ∈ Vertex(X, τ )—the barycenter of ∆, such that C∆ ∈ interior(∆).
(Here interior(∆) is clearly defined by referring ∆ as a simplex of (X, σ).)
Define
U∆ = Star(X,τ ) ({C∆ }).
As in (a), {U∆ | ∆ is a simplex of (X, σ)} is an open cover. In fact,
U∆ ⊃ interior(∆). This open cover is parameterized by simplices of (X, σ)
(also by vertices of (X, τ ), since there is a one to one correspondence between
the vertices of (X, τ ) and the simplices of (X, σ)).
This cover satisfies the following condition: The intersection U∆1 ∩ U∆2 ∩ · · · ∩
U∆k is nonempty if and only if one can reorder the simplices, such that
∆1 ⊂ ∆2 ⊂ · · · ∆n .
One can verify that the open cover in (b) is a refinement of the open cover in
(a).
1.4.3. It is well known that a simplicial complex X is locally contractible.
That is, for any point x ∈ X and an open neighborhood U ∋ x, there is an
open neighborhood W ∋ x with W ⊂ U such that W can be contracted to a
single point inside U . (One can prove this fact directly, using the metric in
1.4.1.)
One can endow different metrics on X, but all the metrics are required to
induce the same topology as the one in 1.4.1.
Using the local contractibility and the compactness of X, one can prove the
following fact.
For any simplicial complex X with a metric d (may be different from the metric
in 1.4.1), there are δX,d > 0 and a nondecreasing function ρ : (0, δX,d ] → IR+
such that the following are true.
(1) limδ→0+ ρ(δ) = 0, and
(2) for any δ ∈ (0, δX,d ] and x0 ∈ X, the ball Bδ (x0 ) with radius δ and centre x0
(see 1.1.7 (d) for the notation) can be contracted into a single point within the
ball Bρ(δ) (x0 ). I.e., there is a continuous map α : Bδ (x0 ) × [0, 1] → Bρ(δ) (x0 )
such that
(i) α(x, 0) = x for any x ∈ Bδ (x0 ),
(ii) α(x, 1) = x0 for any x ∈ Bδ (x0 ).
The following lemma is a consequence of the above fact.
Lemma 1.4.4. For any simplicial complex X with metric d, there are δX,d > 0
and a nondecreasing function ρ : (0, δX,d ] → IR+ such that the following are
true.
Documenta Mathematica 7 (2002) 255–461
276
Guihua Gong
(1) limδ→0+ ρ(δ) = 0, and
(2) for any ball Bδ (x0 ) with radius δ ≤ δX,d , any simplex ∆ (not assumed to
be a simplex in X), and any continuous map f : ∂∆ → Bδ (x0 ), there is a
continuous map g : ∆ → Bρ(δ) (x0 ) such that g(y) = f (y) for any y ∈ ∂∆.
Proof: The simplex ∆ can be identified with ∂∆ × [0, 1]/∂∆ × {1} in such a
way that ∂∆ is identified with ∂∆ × {0}. Define the map g by
g(y, t) = α(f (y), t) ∈ Bρ(δ) (x0 ), ∀y ∈ ∂∆, t ∈ [0, 1],
where α is the map in 1.4.3.
⊔
⊓
1.4.5. The following is a well known result in differential topology: Suppose
that M is an m-dimensional smooth manifold, N ⊂ M is an n-dimensional
submanifold. If Y is an l-dimensional simplicial complex with l < m − n, then
for any continuous map f : Y → M and any ε > 0, there is a continuous map
g : Y → M such that
(i) g(Y ) ∩ N = ∅ and
(ii) dist(g(y), f (y)) < ε, for any y ∈ Y .
There is an analogous result in the case of a simplicial complex M and a subcomplex N . Instead of the assumption that M is an m-dimensional smooth
manifold, let us suppose that M has the property D(m): for each x ∈ M ,
there is a contractible open neighborhood Ux ∋ x such that Ux \ {x} is (m − 2)connected, i.e.,
πi (Ux \ {x}) = 0 for any i ∈ {0, 1, · · · m − 2}.
(We use the following convention: by π0 (X) = 0, it will be meant that X is a
path connected nonempty space.)
Note that IRm \{0} is (m − 2)-connected. Therefore, any m-dimensional manifold has property D(m).
The following result is the relative version of Theorem 5.4.16 of [St] (see page
111 of [St]), which also holds according to the top of page 112 of [St].
Proposition 1.4.6. Suppose that M is a simplicial complex with property
D(m), and N ⊂ M is a sub-simplicial complex. Suppose that Y is a simplicial
complex of dimension l < m − dim(N ), and suppose that Y1 ⊂ Y is a subsimplicial complex. Suppose that f : Y → M is a continuous map such that
f (Y1 ) ∩ N = ∅. For any ε > 0, there is a continuous map f1 : Y → M such
that
(i) f1 |Y1 = f |Y1 ,
(ii) f1 (Y ) ∩ N = ∅, and
(iii) d(f (y), f1 (y)) < ε for any y ∈ Y .
1.4.7. Let X, F be two simplicial complexes.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
277
Let Γ ⊂ Homeo(F ) be a subgroup of the group of homeomorphisms of the
space F .
Let us recall the definition of fibre bundle. A fibre bundle over X with
fibre F and structure group Γ, is a simplicial complex M with a continuous surjection p : M → X such that the following is true. There is an open
cover U of X, and associated to each U ∈ U, there is a homeomorphism
tU : p−1 (U ) → U × F
(called a local trivialization of the bundle) such that
(1) Each tU takes the fibre of p−1 (U ) at x ∈ U to the fibre of U × F at the
same point x—a trivialization of the restriction p−1 (U ) of the fibre bundle to
U , i.e., the diagram
tU
p−1 (U )
❅
p❅
❅
❘
❅
✲ U ×F
¡
¡
✠
¡
¡
p1
U
is commutative, where p1 denotes the project map from the product U × F to
the first factor U , and
(2) The given local trivializations differ fibre-wise only by homeomorphisms in
the structure group Γ: for any U, V ∈ U and x ∈ U ∩ V ,
tU ◦ t−1
V |{x}×F ∈ Γ ⊂ Homeo(F ).
Furthermore, we will also suppose that the metric d of F is invariant
under the action of any element g ∈ Γ, i.e., d(g(x), g(y)) = d(x, y) for
any x, y ∈ F . We will see, the fibre bundles constructed in §2, satisfy this
condition.
(Note that, if F is the vector space IRn with Euclidean metric d, then d is
invariant under the action of O(n) ⊂ Homeo(F ), but not invariant under the
action of Gl(n) ⊂ Homeo(F ).)
A subset F1 ⊂ F is called Γ-invariant if for any g ∈ Γ, g(F1 ) ⊂ F1 .
1.4.8. A cross section of fibre bundle p : M → X is a continuous map
f : X → M such that
(p ◦ f )(x) = x for any x ∈ X.
The following Theorem is a consequence of Proposition 1.4.6. The proof is a
standard argument often used in the construction of cross sections for fibre
bundles (see [Wh]). (In the literature such an argument is often taken for
granted.) We give it here for the convenience of the reader.
Documenta Mathematica 7 (2002) 255–461
278
Guihua Gong
Theorem 1.4.9. Suppose that p : M → X is a fibre bundle with fibre F .
Suppose that F1 ⊂ F is a Γ-invariant sub-simplicial complex of F . Suppose
that F has the property D(m) and dim(X) < m − dim(F1 ). Then for any cross
section s : X → M and any ε > 0, there is a cross section s1 : X → M such
that the following two statements are true:
(1) (tU ◦ s1 )(x) ∈
/ {x} × F1 for any x ∈ U ∈ U, where U is the open cover of
X and {tU }U ∈U is the local trivialization of the fibre bundle. That is, s1 (X)
avoids F1 in each fibre;
(2) d(s1 (x), s(x)) < ε for any x ∈ X, where the distance is taken in the fibre F .
Proof: If the fibre bundle is trivial, then the cross sections of the bundle can be
identified with maps from X to F . The conclusion follows immediately from
1.4.6.
For the general case, we will use the local trivializations.
For each open set U ∈ U, using the trivialization
tU : p−1 (U ) → U × F,
each cross section f on U induces a continuous map f˜U : U → F by
f˜U (x) = p2 (tU (f (x))),
where p2 : U × F → F is the projection onto the second factor.
Suppose that δF is as in 1.4.3 (see 1.4.4 also). That is, there is a nondecreasing
function ρ : (0, δF ] → (0, ∞) such that limδ→0+ ρ(δ) = 0 and such that any
δ-ball Bδ (x) can be contracted to a single point within Bρ(δ) (x).
Let dim(X) = n. Choose a finite sequence of positive numbers
εn > εn−1 > εn−2 > · · · > ε1 > ε0 > 0
as follows. Set εn = min{δF , ε}. Then choose εn−1 to satisfy
ρ(3εn−1 ) <
1
εn
3
and 3εn−1 <
1
εn .
3
Once εl is defined, then choose εl−1 to satisfy
ρ(3εl−1 ) <
1
1
εl and 3εl−1 < εl .
3
3
Repeat this procedure until we choose ε0 to satisfy
ρ(3ε0 ) <
1
1
ε1 and 3ε0 < ε1 .
3
3
Let us refine the given simplicial complex structure on X in such a way that
each simplex ∆ is covered by an open set U ∈ U and that for any simplex ∆
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
279
and an open set U ∈ U which covers ∆, the map s̃U |∆ : ∆ → F , induced by
the cross section s, satisfies
diameter(s̃U (∆)) < ε0 .
(Since the metric on F is invariant under the action of any element in Γ, the
above inequality holds or not does not depend on the choice of the open set
U which covers ∆. In what follows, we will use this fact many times without
saying so.)
We will apply Proposition 1.4.6 to each simplex of X from the lowest dimension
to the highest dimension.
For any l ∈ {0, 1, · · · , n}, let us denote the l-skeleton of X by X (l) . So X (n) =
X, and X (0) is the set of vertices of X.
Step 1. Fix a vertex x ∈ X (0) , and suppose that x ∈ U ∈ U. Applying
Proposition 1.4.6 to {x} (in place of Y with Y1 = ∅) and F ( in place of M
with N = F1 ), there exists s̃0 (x) ∈ F \ F1 such that
d(s̃0 (x), s̃U (x)) < ε0 .
Any choice of s̃0 (x) gives a cross section s0 on {x} by
0
s0 (x) = t−1
U (x, s̃ (x)),
where (x, s̃0 (x)) ∈ {x} × F ⊂ U × F. Defining s0 on all vertices, we obtain a
cross section s0 on X (0) such that
d(s̃0U (x), s̃U (x)) < ε0
for each x ∈ U ∩ X (0) .
Step 2. Suppose that for l < n = dim(X), there is a cross section sl : X (l) →
M such that for any U ∈ U and any x ∈ X (l) ∩ U , we have s̃lU (x) ∈
/ F1 , and
(∗)
d(s̃lU (x), s̃U (x)) < εl .
Let us define a cross section sl+1 : X (l+1) → M as follows. We will work one
by one on each (l + 1)-simplex ∆.
First, we shall simply extend the cross section sl |∂∆ to a cross section on ∆ (see
Substep 2.1 below). Then, apply Proposition 1.4.6 to perturb the cross section
sl |∆ to avoid F1 in each fibre (see Substep 2.2 below). Again, since Proposition
1.4.6 is only for maps (not for cross sections), we will use s̃lU |∂∆ : ∂∆ → F to
replace sl |∂∆, as in Step 1.
Substep 2.1. Let ∆ be an (l + 1)-simplex. Suppose that ∆ ⊂ U ∈ U. Then
s̃lU |∂∆ : ∂∆ → F is a continuous map. Since (∗) holds for any x ∈ ∂∆, and
since
diameter(s̃U (∆)) < ε0 ,
we have
diameter(s̃lU (∂∆)) < εl + εl + ε0 .
Documenta Mathematica 7 (2002) 255–461
280
Guihua Gong
Let δ = εl + εl + ε0 < δF . Then there is a y ∈ F such that s̃lU (∂∆) ⊂ Bδ (y).
Since ρ(δ) ≤ ρ(3εl ) < 31 εl+1 , by Lemma 1.4.4, s̃lU : ∂∆ → F can be extended
to a map (still denoted by s̃lU )
s̃lU : ∆ → F,
such that s̃lU (∆) ⊂ B 31 εl+1 (y). Consequently, the extended map s̃lU also satisfies
that diameter(s̃lU (∆)) < 23 εl+1 .
Substep 2.2. Note that s̃lU (x) ∈
/ F1 for any x ∈ ∂∆. Applying Proposition
1.4.6 to ∆ (in place of Y with subcomplex Y1 = ∂∆) and to F (in the place of
M with subcomplex N = F1 ), we obtain a continuous map
s̃l+1 : ∆ → F
such that
(1) s̃l+1 (x) ∈
/ F1 for any x ∈ ∆,
(2) d(s̃l+1 (x), s̃lU (x)) < ε0 , for any x ∈ ∆, and
(3) s̃l+1 |∂∆ = s̃lU |∂∆.
The map s̃l+1 defines a cross section sl+1 by
l+1
sl+1 (x) = t−1
(x)).
U (x, s̃
After working out all the (l + 1)-simplices, we obtain a cross section sl+1 on
X (l+1) —it is a continuous cross section because it is continuous on each (l + 1)simplex and sl+1 |∂∆ = sl |∂∆ from (3) above.
Recall that diameter(s̃U (∆)) < ε0 and diameter(s̃lU (∆)) < 32 εl+1 . Combining
these facts with (∗), we have
(∗∗)
2
d(s̃lU (x), s̃U (x)) < εl + εl+1 + ε0
3
for any x ∈ ∆. Combining (∗∗) and (2) above, we have
2
d(s̃l+1
U (x), s̃U (x)) < εl + εl+1 + 2ε0 < εl+1
3
for any x ∈ X l+1 ∩ U . This is (∗) for l + 1 (in place of l).
Step 3. By mathematical induction, we can define sl for each l = 0, 1, · · · n as
the above. Let s1 = sn to finish the proof.
⊔
⊓
The following relative version of the theorem is also true.
Corollary 1.4.10. Suppose that p : M → X is a fibre bundle with fibre F
and F1 is a Γ-invariant sub-simplicial complex of F . Suppose that F has the
property D(m) and that dim(X) < m − dim(F1 ). Suppose that X1 ⊂ X is a
sub-simplicial complex. Suppose that the cross section s : X → M satisfies that
(tU ◦ s)(x) ∈
/ {x} × F1 for any U ∈ U and any x ∈ X1 ∩ U , where U and tU
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
281
are as in the definition of fibre bundle in 1.4.7. Then for any ε > 0, there is a
cross section s1 : X → M such that the following three statements are true:
(1) (tU ◦ s1 )(x) ∈
/ {x} × F1 , for any x ∈ U and U ∈ U;
(2) d(s1 (x), s(x)) < ε for any x ∈ X, where the distance is taken inside the
fibre F ;
(3) s1 |X1 = s|X1 .
Proof: In the proof of Theorem 1.4.9, we have essentially proved this relative
version. If fact, in Step 2, we proved that a cross section on a simplex ∆ can be
constructed within arbitrarily small distance of the original cross section such
that
(1) it avoids F1 in each fibre, and
(2) it agrees with the original cross section on ∂∆, provided that the original
cross section avoids F1 on ∂∆.
This is a local version of the Corollary. To prove the Corollary, one only needs
to apply this local version, repeatedly, to the simplices ∆ with ∆\∂∆ ⊂ X\X 1 ,
from the lowest dimension to the highest dimension.
⊔
⊓
1.5
About the decomposition theorem
In this subsection, we will briefly discuss the main ideas in the proof of the
decomposition theorem stated in §4. Mainly, we will review the ideas in the
proofs of special cases already in the literature (see especially [EG2, Theorem
2.21]), point out the additional difficulties in our new setting, and discuss how
to overcome these difficulties. This subsection could be skipped without any
logical gap, but we do not encourage the reader to do so, except for the expert
in the classification theory. By reading this subsection, the reader will get the
overall picture of the proof. In particular, how §2, §3, and the results of [Li2] fit
into the picture. We will also discuss some ideas in the proof of the combinatorial results of §3. This subsection may also be helpful for understanding the
corresponding parts of [EG2], [Li3], and (perhaps) other papers. Even though
the discussion in this subsection is sketched, the proof of Lemma 1.5.4 and
Propositions 1.5.7 and 1.5.7’ are complete. We will begin our discussion with
some very elementary facts.
1.5.1. Let A and B be unital C ∗ -algebras, and φ : A → B, a unital homomorphism. If P ∈ B is a projection which commutes with the image of φ, i.e., such
that
P φ(a) − φ(a)P = 0, ∀a ∈ A,
then φ(a) can be decomposed into two mutually orthogonal parts φ(a)P =
P φ(a)P and φ(a)(1 − P ) = (1 − P )φ(a)(1 − P ):
φ(a) = P φ(a)P + (1 − P )φ(a)(1 − P ).
Documenta Mathematica 7 (2002) 255–461
282
Guihua Gong
1.5.2. In 1.5.1, let us consider the case that A = C(X). Let F ⊂ C(X) be
a finite set. Let unital homomorphism φ : C(X) → B and projection P ∈ B
be as in 1.5.1. Furthermore, suppose that there is a point x0 ∈ X such that
P φ(f )P = φ(f )P is approximately equal to f (x0 )P to within ε on F :
kφ(f )P − f (x0 )P k < ε,
∀f ∈ F.
Then
kφ(f ) − (1 − P )φ(f )(1 − P ) ⊕ f (x0 )P k < ε,
∀f ∈ F.
More generally, if there are mutually orthogonal projections P1 , P2 , · · · , Pn ∈ B,
which commute with φ(C(X)), and points x1 , x2 , · · · , xn ∈ X such that
kφ(f )Pi − f (xi )Pi k < ε,
(∗)
∀f ∈ F, i = 1, 2, · · · n,
then
(∗∗)
kφ(f ) − (1 −
n
X
i=1
Pi )φ(f )(1 −
n
X
i=1
Pi ) ⊕
n
X
i=1
f (xi )Pi k < ε, ∀f ∈ F.
Here, we used the following fact: the norm of the summation of a set of mutually orthogonal elements in a C ∗ -algebra is the maximum of the norms of all
individual elements in the set. In this paper, this fact will be used many times
without saying so.
Example 1.5.3. Let F ⊂ C(X) be a finite set, and ε > 0. Choose η > 0 such
that if dist(x, x′ ) < η, then |f (x) − f (x′ )| < ε for any f ∈ F .
Let
x1 , x2 , · · · , xn
∈
X
be
distinct
points,
and
U1 ∋ x1 , U2 ∋ x2 , · · · , Un ∋ xn be mutually disjoint open neighborhoods
with Ui ⊂ Bη (xi ) (= {x ∈ X | dist(x, xi ) < η}).
| ) and let φ : C(X) → M• (C
| ) be a homoConsider the case that B = M• (C
morphism. If Pi , i = 1, 2, · · · , n are the spectral projections corresponding to
the open sets Ui (see Definition 1.2.4), then the projections Pi commute with
φ(C(X)) and satisfy (*) in 1.5.2. Therefore, the decomposition
(∗∗)
kφ(f ) − (1 −
n
X
i=1
Pi )φ(f )(1 −
n
X
i=1
Pi ) ⊕
n
X
i=1
f (xi )Pi k < ε,
holds for all f ∈ F .
We remark that if #(SPφ ∩ Ui ) (counting multiplicities) is large, then, in the
decomposition, rank(Pi )(= #(SPφ ∩ Ui )) is large.
In the setting of 1.5.2, not only is (**) true for the original projections P1 , P2 , · · · , Pn , but also it is true for any subprojections
p1 ≤ P1 , p2 ≤ P2 , · · · , pn ≤ Pn , with ε replaced by 3ε. Namely, the following lemma holds.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
283
Lemma 1.5.4. Let X be a compact metrizable space, and write A = C(X).
Let F ⊂ A be a finite set. Let B be a unital C ∗ -algebra, and φ : A → B
be a homomorphism. Let ε > 0. Suppose that there are mutually orthogonal projections P1 , P2 , · · · , Pn in B and points x1 , x2 , · · · , xn in X such that
Pi φ(f ) = φ(f )Pi (i = 1, 2, · · · , n) for any f ∈ C(X) and such that
(∗)
kφ(f )Pi − f (xi )Pi k < ε
(i = 1, 2, · · · , n)
for any f ∈ F.
If p1 , p2 , · · · , pn are subprojections of P1 , P2 , · · · , Pn respectively, then
kφ(f ) − (1 −
n
X
i=1
pi )φ(f )(1 −
n
X
i=1
pi ) ⊕
n
X
i=1
f (xi )pi k < 3ε,
for any f ∈ F .
(Notice that the condition that the projections Pi commute with φ(f ) does not
by itself imply that the pi almost commute with φ(f ), but this does follows if
(*) holds.)
Different versions of this lemma have appeared in a number of papers (especially, [Cu], [GL], [EGLP]).
Proof: The proof is a straightforward calculation.
One verifies directly that
X
X
X
k(
Pi )φ(f )(
Pi ) −
f (xi )Pi k < ε, ∀f ∈ F.
P
P
Hence on multiplying by 1 − pi and
pi (one on each side),
X
X
X
X
k(1 −
pi )φ(f )(
pi )k < ε, and k(
pi )φ(f )(1 −
pi )k < ε, ∀f ∈ F ;
P
on multiplying by
pi on both sides,
X
X
X
k(
pi )φ(f )(
pi ) −
f (xi )pi k < ε, ∀f ∈ F.
The desired conclusion follows from identity
X
X
X
X
φ(f ) = ((1 −
pi ) +
pi )φ(f )((1 −
pi ) +
pi ).
⊔
⊓
Remark 1.5.5. One may wonder why we need the decomposition given in the
preceding lemma. In fact, the decomposition (**) of 1.5.2, with the original
projections, has a better estimation. Why do we need to use subprojections?
The reason is as follows.
Suppose that the C ∗ -algebra A = C(X), the finite set F ⊂ A, the points
x1 , x2 , · · · , xn ∈ X, and the open sets U1 ∋ x1 , U2 ∋ x2 , · · · , Un ∋ xn are as in
| ) in 1.5.3),
1.5.3. Let us consider the case B = M• (C(Y )) (instead of M• (C
where Y is a simplicial complex.
Documenta Mathematica 7 (2002) 255–461
284
Guihua Gong
Let φ : C(X) → M• (C(Y )) be a unital homomorphism. As in 1.5.3, let Pi (y)
denote the spectral projection of φ|y corresponding to the open set Ui (see
1.2.8). Then for each y ∈ Y , we have the inequality (**) above,
kφ(f )(y) − (1 −
n
X
i=1
Pi (y))φ(f )(y)(1 −
n
X
i=1
Pi (y)) ⊕
n
X
i=1
f (xi )Pi (y)k < ε, ∀f ∈ F.
Unfortunately, Pi (y) does not in general depend continuously on y, and so this
estimation does not give rise to a decomposition for φ globally.
On the other hand, one can construct a globally defined continuous projection pi (y) which is a subprojection of Pi (y) at each point y, and is such
that rank(pi ) is not much smaller than miny∈Y rank(Pi (y)) (more precisely,
rank(pi ) ≥ miny∈Y rank(Pi (y)) − dim(Y )), by using the continuous selection
theorem of [DNNP] as 1.5.6 below.
Once this is done , then for each y ∈ Y , applying the lemma, we have
kφ(f )(y) − (1 −
n
X
i=1
pi (y))φ(f )(y)(1 −
n
X
i=1
pi (y)) ⊕
n
X
i=1
f (xi )pi (y)k < 3ε, ∀f ∈ F.
Since the projections pi (y) depend continuously on y, they define elements
pi ∈ B. We can then rewrite the preceding estimate as
kφ(f ) − (1 −
n
X
i=1
pi )φ(f )(1 −
n
X
i=1
pi ) ⊕
n
X
i=1
f (xi )pi k < 3ε, ∀f ∈ F.
1.5.6. We would like to discuss how to construct the projections pi referred to
in 1.5.5, using the selection theorem [DNNP 3.2].
To guarantee pi to have a large rank, we should assume that Pi (y) has a large
rank at every point y. So let us assume that for some positive integer ki and
for every point y ∈ Y ,
#(SPφy ∩ Ui ) ≥ ki ,
equivalently, rank(Pi (y)) ≥ ki .
For the sake of simplicity, let us fix i and write U for Ui (U ⊂ X), P for Pi , k
for ki , and p for the desired projection pi . So for every point y ∈ Y ,
#(SPφy ∩ U ) ≥ k,
equivalently, rankP (y) ≥ k. Let us construct a projection p(y), depending
continuously on y, such that rankp(y) ≥ k − dim(Y ) and p(y) ≤ P (y) for each
y ∈Y.
For each fixed y0 ∈ Y , since SPφy0 ∩ U is a finite set, one can choose an
open set U ′ ⊂ U ′ ⊂ U such that SPφy0 ∩ U ′ = SPφy0 ∩ U . In particular,
SPφy0 ∩ (U ′ \U ′ ) = ∅. By Lemma 1.2.10, there is a connected open set W ∋ y0
in Y such that
SPφy ∩ (U ′ \U ′ ) = ∅, ∀y ∈ W.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
285
Let P W (y) be the spectral projection of φy corresponding to open set U ′ . By
Lemma 1.2.9, this depends continuously on y, and so defines a continuous
projection-valued function
| ).
P W : W → projections of M• (C
Furthermore, P W (y) ≤ P (y) for any y ∈ Y and, for each y in the (connected)
subset W ,
rank(P W (y)) = #(SPφy0 ∩ U ′ ) = #(SPφy0 ∩ U ) ≥ k.
Once we have the above locally defined continuous projection-valued functions
P W (y), the existence of a globally defined continuous projection-valued function p(y) follows from the following result.
Proposition ([DNNP 3.2]). Let Y be a simplicial complex, and let k be a
positive integer. Suppose that W is an open covering of Y such that for each
| )
W ∈ W, there is a continuous projection-valued map P W : W → M• (C
satisfying
rankP W (y) ≥ k
for all y ∈ W.
| ) such that
Then there is a continuous projection-valued map p : Y → M• (C
for each y ∈ Y ,
rank p(y) ≥ k − dim(Y ), and
p(y) ≤
_
{P W (y);
W ∈ W, y ∈ W }.
Let p(y) be as given in the preceding proposition with respect to P W as defined
above. Then as P W (y) ≤ P (y) for each W , p(y) ≤ P (y) also holds .
Recall, we write U for Ui , P for Pi and p for pi . So, we obtain a projection
pi such that pi (y) is a subprojection of Pi (y) for every y. Since Pi (y), i =
1, 2, · · · , n, are the spectral projections corresponding to Ui , i = 1, 2, · · · n,
which are mutually disjoint, the projections Pi (y), i = 1, 2, · · · , n, are mutually
orthogonal, and so are the projections pi , i = 1, 2, · · · , n. Combining this
construction with Lemma 1.5.4, we have the following result.
Proposition 1.5.7.
Let X be a simplicial complex, and F ⊂ C(X) a finite subset. Suppose that ε > 0 and η > 0 are as in 1.5.3, i.e., such that if
dist(x, x′ ) < η, then |f (x) − f (x′ )| < ε for any f ∈ F .
Suppose that U1 , U2 , · · · , Un are disjoint open neighborhoods of (distinct) points
x1 , x2 , · · · , xn ∈ X, respectively, such that Ui ⊂ Bη (xi ) for all 1 ≤ i ≤ n.
Suppose that φ : C(X) → M• (C(Y )) is a unital homomorphism, where Y is a
simplicial complex, such that
#(SPφy ∩ Ui ) ≥ ki
for 1 ≤ i ≤ n, and for all y ∈ Y.
Documenta Mathematica 7 (2002) 255–461
286
Guihua Gong
Then there are mutually orthogonal projections p1 , p2 , · · · , pn ∈ M• (C(Y )) with
rank(pi ) ≥ ki − dim(Y ) such that
kφ(f ) − p0 φ(f )p0 ⊕
where p0 = 1 −
P
n
X
i=1
f (xi )pi k < 3ε
for all f ∈ F,
pi . Consequently,
rank(p0 ) ≤ (#(SPφy ) −
n
X
i=1
ki ) + n · dim(Y ).
(Note that #(SPφy ) is the order of the matrix algebra M• (C(Y )).)
(In fact, the above is also true if one replaces M• (C(Y )) by P M• (C(Y ))P ,
with the exact same proof.)
1.5.8. Proposition 1.5.7 is implicitly contained in the proof of the main decomposition theorem—Theorem 2.21 of [EG2], and explicitly stated as Theorem
2.3 of [Li3], for the case of dimension one.
To use 1.5.7 to decompose a partial map φi,j
M[m,i] (C(Xm,i )) →
m,m′ :
M[m′ ,j] (C(Xm′ ,j )) of the connecting homomorphism φm,m′ : Am → Am′ in
the inductive system (Am , φm,m′ ), we only need to write
φi,j
m,m′ = φ ⊗ id[m,i] ,
(see 1.2.9), and then decompose φ (cf. 1.2.19). In [EG2], we proved that such a
map φ (for m′ large enough) satisfies the condition in Proposition 1.5.7, if the
inductive limit is of real rank zero. More precisely, we constructed
Pn mutually
disjoint open sets U1 , U2 , · · · , Un , with
small
diameter,
such
that
i=1 ki is very
Pn
large compared with (#(SPφy ) − i=1 ki ), where ki = miny∈Y #(SPφy ∩ Ui ).
(See the open sets Wi in the proof ofPTheorem 2.21 of [EG2].) Therefore,
n
inPthe above decomposition, the part
i=1 f (xi )pi , which has rank at least
n
( i=1 ki ) − n · dim(Y ), has much largerPsize than the size of the part p0 φ(f )p0 ,
n
which has rank at most (#(SPφy ) − i=1 ki ) + n · dim(Y ), if n · dim(Y ) is
very small
Pn compared with #(SPφy ). (Notice that if φ is not unital, then p0 =
φ(1)− i=1 pi and #(SPφy ) = rankφ(1).) (We should mention that n·dim(Y )
is automatically small from the construction. This is a kind of technical detail,
to which the reader should not pay much attention now. The number n depends
only on η above, but #(SPφy ) could be very large as m′ (for φm,m′ ) is large.
In particular, it could be much larger than dim(Y ) (note that Y = Xm′ ,j ), if
the inductive limit has slow dimension growth.)
The above construction is not trivial. It depends heavily on the real rank zero
property and Su’s result concerning spectral variation (see [Su]).
What was proved by this construction in [EG2] is the decomposition theorem
for the real rank zero case, as mentioned in the introduction.
1.5.9. For the case of a non real rank zero inductive system, we can not
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
287
construct the mutually disjoint open set {Ui } as described in 1.5.8. Notice
that in the
Pn decomposition described in 1.5.8, the major part ψ defined by
ψ(f ) := i=1 f (xi )pi has property that
∼rank(p1 )
SPψy = {x1
∼rank(p2 )
, x2
, · · · , xn∼rank(pn ) }.
That is, the spectrum consists of several fixed points {xi }ni=1 (⊂ X) with multiplicities. This kind of decomposition depends on the real rank zero property.
For the decomposition of the simple inductive limit algebras, we are forced to
allow the major part to have variable spectrum— SPψy varies when y varies.
The following results can be proved exactly the same as the way Proposition
1.5.7 is proved (see 1.5.6 and 1.5.4). (Proposition 1.5.7 is a special case of the
following result by taking αi (y) = xi , the constant maps, and Ui (y) = Ui ∋ xi ,
the fixed open sets.)
Proposition 1.5.7’.
Let X be a simplicial complex, and F ⊂ C(X), a
finite subset. Suppose that ε > 0 and η > 0 are as in 1.5.3, i.e., such that if
dist(x, x′ ) < η, then |f (x) − f (x′ )| < ε for any f ∈ F .
Suppose that α1 , α2 , · · · αn : Y → X are continuous maps from a simplicial
complex Y to X. Suppose that U1 (y), U2 (y), · · · , Un (y) are mutually disjoint
open sets satisfying Ui ⊂ Bη (αi (y)) and satisfying the following continuity
condition:
For any y0 ∈ Y and closed set F ⊂ Ui (y0 ), there is an open set W ∋ y0 such
that F ⊂ Ui (y) for any y ∈ W .
Suppose that φ : C(X) → M• (C(Y )) is a unital homomorphism such that
#(SPφy
\
Ui (y)) ≥ ki ,
∀i ∈ {1, 2, · · · , n}, y ∈ Y.
Then there are mutually orthogonal projections p1 , p2 , · · · , pn ∈ M• (C(Y )) with
rank(pi ) ≥ ki − dim(Y ) such that
kφ(f )(y) − p0 (y)φ(f )(y)p0 (y) ⊕
where p0 = 1 −
P
n
X
f (αi (y))pi (y)k < 3ε,
i=1
∀f ∈ F,
pi . Consequently,
rank(p0 ) ≤ (#(SPφy ) −
n
X
ki ) + n dim(Y ).
i=1
(It is easy to see that the proof of Proposition 1.5.7 (see 1.5.6) can be generalized
to this case. Notice that the above continuity condition for Ui (y) assures that
U ′ ⊂ Ui (y) for any y ∈ W , where U ′ is described in 1.5.6 corresponding to y0
and U = Ui (y0 ). Then it will assure that P W (y) ≤ P (y), where P W (y) is the
locally defined continuous projection-valued function described in 1.5.6.)
Documenta Mathematica 7 (2002) 255–461
288
Guihua Gong
1.5.10. In the above proposition, if one can choose the maps αi : Y → X
factoring through the interval [0, 1]—the best possible 1-dimensional space—as
a
b
i
[0, 1] −→ X,
αi : Y −→
Pn
then the part ψ, defined by ψ(f )(y) =
i=1 f (αi (y))pi (y), factors through
C[0, 1] as
n
n
M
M
ψ′
b∗
pi ),
pi )M• (C(Y ))(
C(X) −→ C([0, 1]) −→ (
i=1
i=1
∗
′
where b is induced by b : [0, 1] → X and ψ is defined by
ψ ′ (f )(y) =
n
X
f (ai (y))pi (y).
i=1
In particular, if
(1)
n
X
i=1
ki >> (#(SPφy ) −
n
X
ki ),
i=1
where ki = miny∈Y #(SPφy ∩ Ui ), then we obtain a decomposition with major
part factoring through the interval algebras—direct sum of matrix algebras
over interval.
1.5.11. The ideal approach for obtaining a decomposition of φm,m′ with major part factoring through an interval algebra, is to reduce it to the setting
of Proposition 1.5.7’, that is, to construct continuous maps {αi } (factoring
through interval) and the mutually disjoint open sets {Ui } as described in
1.5.7’, such that (1) in 1.5.10 holds for homomorphism φ induced from the
partial connecting homomorphism φi,j
m,m′ described in 1.5.8.
Unfortunately, it seems impossible to realize such a construction globally.
A consequence of the property of αi described in Proposition 1.5.7’, is the
following property of αi , called property (Pairing):
Property (Pairing): For each y ∈ Y , there is a subset of SPφy , which can
be paired with
{α1 (y)∼k1 , α2 (y)∼k2 , · · · , αn (y)∼kn }
to within η, counting multiplicities. (See 1.1.7 (b) for the notation x∼k .)
Even though one can not construct the continuous maps {αi } (factoring
through interval [0, 1] and open sets {Ui } to satisfy the conditions in Proposition
1.5.7’ together with the condition (1) in 1.5.10, for the connecting homomorphisms in the simple inductive limit, Li constructed the maps {αi } to satisfy
the above weaker property (Pairing) and (1) in 1.5.10.
In fact, Li proves the following lemma.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
289
Lemma. Suppose that lim(Am , φm,m′ ) is a simple AH-inductive limit with slow
dimension growth and with injective connecting homomorphisms. For any η >
0, and Am , there are a δ > 0 and an integer N > m, such that for any m′ > N ,
SP(φi,j
m,m′ )y can be paired with
Θ(y) = {α1 (y)∼T1 , α2 (y)∼T2 , · · · , αL (y)∼TL }
to within η, counting multiplicities, for certain continuous maps αi : Y → X
factoring through [0,1], where X = Xm,i and Y = Xm′ ,j . Furthermore, if
(1Ai ))
rank(φi,j
m,m′
m
by K, then Ti ≥ int(δK) and therefore L < 2δ ,
we denote
rank(1Aim )
provided δK ≥ 2, where int(δK) is the integer part of δK as in 1.1.7.
(This lemma is Theorem 2.19 (see also Remark 2.21) of [Li2]. The additional
part about the size of Ti could be obtained by inspecting the proof of the
theorem (see 2.16 and 2.18 of [Li2]).)
i,j
(Note that when we write φi,j
m,m′ = φ ⊗ id[m,i] , we have SP(φm,m′ )y = SPφy .)
In particular, the condition (1) in 1.5.10 holds, since the right hand side is zero.
Therefore the following theorem will be useful for our setting, which is the first
theorem in §4.
Theorem 4.1. Let X be a connected finite simplicial complex, and F ⊂ C(X)
be a finite set which generates C(X). For any ε > 0, there is an η > 0 such
that the following statement is true.
Suppose that a unital homomorphism φ : C(X) → P M• (C(Y ))P (rank(P ) =
K) (where Y is a finite simplicial complex) satisfies the following condition:
There are L continuous maps
a1 , a2 , · · · , aL : Y −→ X
such that for each y ∈ Y, SPφy and Θ(y) can be paired within η, where
Θ(y) = {a1 (y)∼T1 , a2 (y)∼T2 , · · · , aL (y)∼TL }
and T1 , T2 , · · · , TL are positive integers with
T1 + T2 + · · · + TL = K = rank(P ).
Let T = 2L (dim X + dim Y )3 . It follows that there are L mutually orthogonal
projections p1 , p2 , · · · , pL ∈ P M• (C(Y ))P such that
PL
(i) kφ(f )(y) − p0 (y)φ(f )(y)p0 (y) ⊕ i=1 f (ai (y))pi (y)k < ε for any f ∈ F and
PL
y ∈ Y , where p0 = P − i=1 pi ;
(ii) kp0 (y)φ(f )(y) − φ(f )(y)p0 (y)k < ε for any f ∈ F and y ∈ Y ;
(iii) rank(pi ) ≥ Ti − T for 1 ≤ i ≤ L, and hence rank(p0 ) ≤ LT .
(In the above, η can be chosen to be any number satisfying that if dist(x, x ′ ) <
2η, then |f (x) − f (x′ )| < 3ε , ∀f ∈ F.)
(Note that we can not make the number T in the above theorem as small as
dim(Y ), as in Proposition 1.5.7 or 1.5.7’, for some technical difficulties. This
Documenta Mathematica 7 (2002) 255–461
290
Guihua Gong
is the reason that we are forced to use the stronger condition of very slow
dimension growth instead of slow dimension growth in our main decomposition
theorem.)
1.5.12. The proof of the above theorem is much more difficult than that of
Proposition 1.5.7 or 1.5.7’. In particular, Theorem 2.1 (see §1.2 above), and
results in §3 are only for the purpose of proving the above theorem. With
these results in hand, the proof of Theorem 4.1 will be given in 4.2–4.19. Then
the main decomposition theorem described in the introduction—Theorems 4.35
and 4.37—will be proved based on Theorem 4.1 and the results from [Li2].
We would like to explain the difficulties, and how Theorem 2.1 and §3 will be
used to over come the difficulties.
Now, our notations are as in Theorem 4.1 above.
Fix i. Let Ui (y) = {x ∈ X | dist(x, ai (y)) < η}, then from the condition of
Theorem 4.1, we have
#(SPφy ∩ Ui (y)) ≥ Ti .
Let Pi (y) be the spectral projection corresponding to the open set Ui (y). This
is not a continuously defined projection. But using the same procedure in 1.5.6
(see 1.5.7 and 1.5.7’), one can construct a globally defined projection pi (y) such
that pi (y) ≤ Pi (y), and rank(pi (y)) ≥ Ti − dim(Y ).
But unfortunately, those pi (y) are not mutually orthogonal, since Ui (y) are not
mutually disjoint, and therefore Pi (y) are not mutually orthogonal.
1.5.13. If we assume that the maximum spectral multiplicity of φ is at most
Ω, then for each y ∈ Y , we can divide the set SPφy (with multiplicity)
into L mutually disjoint subsets E1 , E2 , · · · , EL such that, for each λ ∈ Ei ,
dist(λ, ai (y)) ≤ η, i = 1, 2, · · · , L, and such that
Ti − Ω < #(Ei ) < Ti + Ω, i = 1, 2, · · · , L,
counting multiplicity. By {Ei } being mutually disjoint, we mean that if an
element λ ∈ SPφy has multiplicity k, then we put the entire k copies of λ into
one of Ei , without separating them. (In the above, φ, ai , Ti , and L are all
from Theorem 4.1.)
(Note that if we require that #(Ei ) = Ti , then we can not guarantee {Ei } are
mutually disjoint, because of spectral multiplicity.)
Then we can construct mutually disjoint open sets U1 (y), U2 (y), · · · , UL (y) such
that Ei ⊂ Ui (y) and Ui (y) ⊂ Bη (ai (y)). We can further assume that these
open sets have mutually disjoint closures. That is Ui (y) ∩ Uj (y) = ∅, for
i 6= j, i, j ∈ {1, 2, · · · , L}.
(The open sets from such construction usually do not satisfy the continuity
condition in Proposition 1.5.7’, so we can not apply Proposition 1.5.7’. We
need to check the proof of it (e.g. the argument in 1.5.6) against our new
setting.)
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
291
For each y0 ∈ Y , there is an open set W (y0 ) ∋ y0 such that
SPφy ⊂ U1 (y0 ) ∪ U2 (y0 ) ∪ · · · ∪ UL (y0 ), ∀y ∈ W (y0 ).
As in 1.5.6, one can construct the mutually orthogonal locally defined continuous projection-valued functions
W (y0 )
Pi
| ), i = 1, 2, · · · , L,
: W (y0 ) → projection of M• (C
W (y )
where Pi 0 (y) (y ∈ W (y0 )) are the spectral projections of φy corresponding
W (y )
to open sets Ui (y0 ) (or SPφy ∩ Ui (y0 )). Furthermore, rankPi 0 = #(Ei ) >
Ti − Ω.
(Note that we do not need to introduce the smaller open set U ′ as in 1.5.6,
because it is automatically true that SPφy ∩(Ui (y0 )\Ui (y0 )) = ∅, as {Ui (y0 )}Li=1
are mutually disjoint and SPφy ⊂ U1 (y0 ) ∪ U2 (y0 ) ∪ · · · ∪ UL (y0 ).)
Theorem 2.1 guarantees that Ω is controlled, it is at most dim(X)+
dim(Y ), which will be very small compared with Ti , in our future
application.
1.5.14. There is a finite subcover W = {W (yj )}j of the open cover {W (y)}y∈Y
of Y .
We can use the selection theorem [DNNP 3.2] (see 1.5.6 above) to construct
global defined continuous projection valued functions pi (y), i = 1, 2, · · · , L, of
ranks at least Ti − Ω − dim(Y ), such that
_ W (y )
(∗)
pi (y) ≤ {Pi j (y) | y ∈ W (yj ) and W (yj ) ∈ W}.
For any i1 6= i2 ∈ {1, 2, · · · , L}, W (yj ) ∈ W, and y ∈ W (yj ), we have
W (y )
W (y )
Pi1 j (y)⊥Pi2 j (y).
Unfortunately, when y ∈ W (yj1 ) ∩ W (yj2 ), we do not have
W (yj1 )
Pi1
W (yj2 )
(y)⊥Pi2
(y).
Therefore, one can not conclude that pi1 (y)⊥pi2 (y) from the above (*).
W (y )
(Notice that, in the above, Pi 0 (y) is the spectral projection of φy with
respect to the open set Ui (y0 ) (not Ui (y)), and in general, Ui1 (yj1 )∩Ui2 (yj2 ) 6= ∅
if j1 6= j2 . In Propositions 1.5.7 and 1.5.7’, we do not have such problem, since
PiW (y) is the spectral projection of U ′ , which is an open subset of Ui (does not
depend on y) in the case of Proposition 1.5.7, or which is an open subset of
Ui (y0 ) ∩ Ui (y) in the case of Proposition 1.5.7’; see 1.5.6 and the explanation
after Proposition 1.5.7’ for more details.)
1.5.15. For any W ∈ W and y ∈ W , define QW
i (y), i = 1, 2, · · · , L, to
be
the
spectral
projections
of
φ
at
point
y
with
respect to the open sets
T
U
(y
),
i
=
1,
2,
·
·
·
,
L.
These
are
subprojections
of PiW (y).
{j: W (yj )∩W 6=∅} i j
Documenta Mathematica 7 (2002) 255–461
292
Guihua Gong
The advantage of using these projections is the following fact.
y ∈ W (yj1 ) ∩ W (yj2 ), we do have
W (yj1 )
Qi1
W (yj2 )
(y)⊥Qi2
For any
(y)
for
T any i1 6= i2 ∈ {1, 2, · · · , L}, because Ui1 (yj1 )T∩ Ui2 (yj1 ) = ∅,
{j: W (yj )∩W (yj2 )6=∅} Ui2 (yj ) ⊂
{j: W (yj )∩W (yj1 )6=∅} Ui1 (yj ) ⊂ Ui1 (yj1 ) and
Ui2 (yj1 ) (the second inclusion follows from W (yj1 ) ∩ W (yj2 ) 6= ∅).
W
Then we apply the selection theorem to QW
i (instead of Pi ) to find globally
defined continuous projection-valued functions pi (y) such that
_ W (y )
pi (y) ≤ {Qi j (y) | y ∈ W (yj ) and W (yj ) ∈ W},
and such that
rank(pi (y)) ≥
min {rank(QW
i (y))} − dim(Y ).
y∈W ∈W
(The readers may notice that QW
i (y) are not continuous on W , so one can not
apply the selection
theorem
directly.
But one can introduce the open subsets
T
U ′ as in 1.5.6 for {j: W (yj )∩W 6=∅} Ui (yj ) (instead of Ui ). We omit the details.)
This time, pi1 (y)⊥pi2 (y) for any i1 6= i2 ∈ {1, 2, · · · , L}.
To guarantee rank(pi (y)) to be large—not too much smaller than Ti ,
miny∈W ∈W {rank(QW
i (y))} must be large.
1.5.16. Fixed y0 ∈ Y with W (y0 ) ∈ W. Recall from the definitions of Ui (y0 )
and W (y0 ) (see 1.5.13),
¡
¢ ¡
¢
¡
¢
SPφy = SPφy ∩ U1 (y0 ) ∪ SPφy ∩ U2 (y0 ) ∪· · ·∪ SPφy ∩ UL (y0 ) , ∀y ∈ W (y0 ).
W (y )
W (y )
Define Ei 0 (y) := SPφy ∩ Ui (y0 ). (Ei 0 (y0 ) is the set Ei in 1.5.13, with
y0 in place of y.) Then for each y ∈ W ∈ W, {EiW (y)}Li=1 is a division of SPφy
(in the terminology in §3, it will be called a grouping of SPφy ).
From 1.5.15,
\
rank(QW
Ui (yj )
i (y)) = # SPφy ∩
{j: W (yj )∩W 6=∅}
= #(
\
W (yj )
Ei
(y)).
{j: W (yj )∩W 6=∅}
Roughly speaking, for our construction in 1.5.15 to work, we need the following
condition.
Condition: For each y ∈ Y , the number #(
not too much smaller than Ti .
T
{W : y∈W ∈W}
Documenta Mathematica 7 (2002) 255–461
EiW (y)) is large—
Simple Inductive Limit C ∗ -Algebras, I
293
(This condition is a little weaker than QW
i (y) to be large. But we are going to
use some special open cover so that the above weaker condition will be enough.
We are not going to discuss details here, and the reader does not need to pay
much attention.)
But in 1.5.15, we only have #(EiW (y)) > Ti −Ω. To obtain the above condition,
we need the combinatorial results in §3. We are going to discuss it now.
1.5.17. For an intersection to be large, it will be certainly natural to require
the number of sets involved in the intersection be as small as possible. As in
the setting of 1.5.16, we should require that for any y ∈ Y , the number of sets
in W which cover y—#{W | y ∈ W ∈ W}— is not too large.
From the definition of covering dimension, we know that for any n-dimensional
compact metrizable space Y and any finite cover U of Y , there is a refined cover
U1 of U such that for any point y ∈ Y , there are at most n + 1 open sets in U1
to cover the point y. In particular, for a simplicial complex Y , the construct
of such open cover is given in 1.4.2 (a).
Let {Wy }y∈Vertex(Y ) be the open cover of Y given in 1.4.2(a). Recall that, for
Tk
any vertices y0 , y1 , · · · , yk , the intersection i=1 Wyi is nonempty if and only if
y0 , y1 , · · · , yk share one simplex.
For any finite open cover, there is a finite open cover of the above form (for
some refined simplicial complex structure), refining the given open cover.
Without loss of generality, we can assume the open cover W = {W (yj )} in
1.5.14 is of the above form. Hence {yj } are vertices of the simplicial complex Y
and W (yj ) are the open sets Wyj defined above. Then the condition in 1.5.16
becomes the following.
For any simplex ∆ of Y with vertices y0 , y1 , · · · , yk ,
³
´
W (y )
W (y )
W (y )
# Ei 0 (y) ∩ Ei 1 (y) ∩ · · · ∩ Ei k (y) ≥ Ti − C
for any y ∈ W (y0 ) ∩ W (y1 ) ∩ · · · ∩ W (yk ), where C is not too large.
(In the proof of Theorem 4.1, the number C will be chosen to be
2L Ω(1 + dim(Y )(dim(Y ) + 1)), where Ω is the maximum spectral multiplicity which is bounded by dim(X) + dim(Y ), by Theorem 2.1.)
1.5.18. To make the discussion simpler, we suppose that the homomorphism
φ has distinct spectrum at any point y ∈ Y . That is, the maximum spectral
multiplicity of φ is one. (Of course, in the proof of Theorem 4.1 in §4, we will
not make this assumption.)
If the simplicial structure is sufficiently refined, by the distinct property of
the spectrum,
we can assume the following holds: For any simplex ∆ with
S
Z = y∈Vertex(∆) W (y)( ⊃ ∆), there are continuous maps
λ1 , λ2 , · · · , λK : Z → X,
where K = rank(P ) as in Theorem 4.1, such that
SPφy = {λ1 (y), λ2 (y), · · · , λK (y)}, ∀y ∈ Y.
Documenta Mathematica 7 (2002) 255–461
294
Guihua Gong
W (y)
Then for any y ∈ Vertex(∆), the division (or grouping) Ei
(y) of SPφy gives
rise to a division (or grouping) Ei (y) of {λ1 , λ2 , · · · , λK }. In the case of distinct
spectrum, the condition in 1.5.13 concerning #(Ei ) is #(Ei (y)) = Ti . The
condition at the end of 1.5.17 is
\
#
Ei (y) ≥ Ti − C.
y∈Vertex(∆)
This is of course not true in general, unless we make some special arrangement.
But with the following lemma, we can always subdivide (or refine) the simplicial complex and introduce the groupings of {λ1 , λ2 , · · · , λK } for all newly
introduced vertices, to make the above true for any simplex of new simplicial
structure (after subdivision).
The following formal definition of grouping is in 3.2 of §3.
Definition. Let E = {1, 2, · · · , K} be an index set. Let T1 , T2 , · · · , TL be non
negative integers with
T1 + T2 + · · · + TL = K.
A grouping of E of type (T1 , T2 , · · · , TL ) is a set of L mutually disjoint index
sets E1 , E2 , · · · , EL with
E = E1 ∪ E2 ∪ · · · ∪ E L ,
and #(Ej ) = Tj for each 1 ≤ j ≤ L.
Lemma.
Suppose that (∆, σ) is a simplex, where σ is the standard
simplicial structure of the simplex ∆.
Suppose that for each vertex
x ∈ Vertex(∆, σ), there is a grouping E1 (x), E2 (x), · · · , EL (x) of E
of type (T1 , T2 , · · · , TL ).
It follows that there is a subdivision (∆, τ ) of (∆, σ), and there is an extension
of the definition of the groupings of E for Vertex(∆, σ) to the groupings of E
(of type (T1 , T2 , · · · , TL )) for Vertex(∆, τ ) (⊃ Vertex(∆, σ)) such that:
(1) For each newly introduced vertex x ∈ Vertex(X, τ ),
[
Ej (x) ⊂
Ej (y),
j = 1, 2, · · · , L.
y∈Vertex(∆,σ)
(2) For any simplex ∆1 of (X, τ ) (after subdivision),
#(
\
x∈Vertex(∆1 )
Ej (x)) ≥ Tj −
dim(∆)(dim(∆) + 1)
,
2
j = 1, 2, · · · , L.
1.5.19. Condition (1) above is important for the following reason. In 1.5.13,
when we define Ei as a subset of SPφy , we require that dist(λ, ai (y)) < η for
any λ ∈ Ei . This condition guarantees that the projection PiW in 1.5.13 (or
W
W
QW
i in 1.5.16) satisfies that φ(f )(y)Pi (y) (or φ(f )(y)Qi (y)) is approximately
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
295
equal to f (ai (y))PiW (y) (or f (ai (y))QW
i (y)) to within ε, which is the condition
(*) in Lemma 1.5.4.
We consider the grouping of E as the grouping of the spectral functions
{λ1 , λ2 , · · · , λK } in 1.5.18. Then the condition (1) in the lemma implies the
following fact. If for any vertex y0 ∈ Vertex(∆, σ) and any element k ∈ Ei (y0 ),
we have dist(λk (y), ai (y)) < η, ∀y ∈ ∆, then for any newly introduced
vertex y1 ∈ Vertex(∆, τ ) and any element k ′ ∈ Ei (y1 ) (here Ei (y1 ) is the
set Ei for the newly introduced grouping for the vertex y1 ), we still have
dist(λk′ (y), ai (y)) < η, ∀y ∈ ∆.
1.5.20. In fact, in the proof of Theorem 4.1, we need the relative version of the
result: Suppose that there are a subdivision (∂∆, τ ′ ) of the boundary (∂∆, σ)
and groupings for all vertices in Vertex(∂∆, τ ′ ) (⊃ Vertex(∆, σ)), such that
the above (1) holds for any vertex in Vertex(∂∆, τ ′ ) and such that the above
(2) holds for any simplex ∆1 of (∂∆, τ ′ ) with dim(∂∆) in place of dim(∆).
Then there is a subdivision (∆, τ ) of (∆, σ), and groupings for all vertices
in Vertex(∆, τ ) such that the above (1) and (2) hold and in addition, the
following holds: the restriction of (∆, τ ) onto the boundary ∂∆ is (∂∆, τ ′ ) and
the grouping associated to any vertex in Vertex(∂∆, τ )(= Vertex(∂∆, τ ′ )) is
the same as the old one.
In §3, we will prove the above relative version. In fact, to prove the absolute version will automatically force us to prove the stronger one—the relative
version.
Another complication comes from the multiplicities, since we can not assume
the spectrum to be distinct. In this case, even the definition of grouping needs
to be modified.
1.5.21. To give the readers some feeling about the lemma in 1.5.18, we shall
discuss the special case of dim(∆) = 1. That is, ∆ = [0, 1], the interval.
In this case, we have two groupings for the end points, {E1 (0), E2 (0), · · · EL (0)}
and {E1 (1), E2 (1), · · · EL (1)} of E of type (T1 , T2 , · · · , TL ). Then we need to
introduce a sequence of points
0 = t0 < t1 < t2 < · · · < tn−1 < tn = 1,
(this give rise to a subdivision of ∆ = [0, 1]) and define groupings
{E1 (tj ), E2 (tj ), · · · EL (tj )} for j = 1, 2, · · · n − 1 such that conditions (1) and
(2) in the lemma holds.
The condition (1) in the lemma means
Ei (tj ) ⊂ Ei (0) ∪ Ei (1), ∀i ∈ {1, 2, · · · L}, j ∈ {1, 2, · · · , n − 1}.
The condition (2) in the lemma means
#(Ei (tj ) ∩ Ei (tj+1 )) ≥ Ti − 1,
i.e., for any i, and any pair of adjacent points tj , tj+1 , the set Ei (tj+1 ) differs
from the set Ei (tj ) by at most one element.
Documenta Mathematica 7 (2002) 255–461
296
Guihua Gong
Let us discuss how to make the above (2) hold for E1 . Suppose that E1 (0) and
E1 (1) are different (otherwise, we do not need to do anything for them). We can
modify E1 (0) to obtain E1 (t1 ) as follows. Taking one element λ in E1 (1)\E1 (0)
to replace an element µ in E1 (0)\E1 (1), and define it to be E1 (t1 ). So E1 (t1 )
contains λ but not µ. Since λ ∈
/ E1 (0), it must be in some Ei (0), i > 1. In the
set Ei (0), after we take out λ and put it into E0 (t1 ), Ei has one element less
than it should have, so we can put µ in it, and call it Ei (t1 ). For j 6= 1 or i,
Ej (t1 ) should be the same as Ej (0). In such a way, we construct the grouping
for t1 , which satisfies the above (2) for the pair 0, t1 . Furthermore, compare
to {Ei (0)}i , the new grouping {Ei (t1 )}i is one step closer to the grouping
{Ei (1)}i . Repeating the above construction (e.g., for t1 in place of 0) we can
construct Ei (t2 ) and so on. Finally, we will reach the grouping at the other
end point 1 ∈ [0, 1].
If one does not require the condition (1), the above is the complete proof of
the lemma for the one-dimensional case.
1.5.22. Since we require the condition (1), when we add an element λ ∈
E1 (1)\E1 (0) into E1 (0) to define E1 (t1 ) (as in 1.5.21), we shall carefully choose
the element µ ∈ E1 (0)\E1 (1) to be replaced by λ. In §3, we shall prove the
following assertion: For any λ ∈ E1 (1)\E1 (0), there is µ ∈ E1 (0)\E1 (1) to
satisfy the following condition: let F = (E1 (0)\{µ}) ∪ {λ}; the set E\F can be
grouped into E2′ , E3′ , · · · , EL′ (#(Ei′ ) = Ti ), in such a way that
Ei ⊂ Ei (0) ∪ Ei (1).
(Such an element µ is the element we should choose.) This is Lemma 3.9 with
Ei (0) ∪ Ei (1) in place of Hi .
With the above ideas in mind, it should not be difficult (hopefully) to read
the first part of §3, which does not involve multiplicity. The main step of §3 is
contained in the proof of Lemma 3.11.
1.5.23. In the case with multiplicity, there are two possible ways of proceeding.
1. Define a grouping of
E = {λ1∼w1 , λ2∼w2 , · · · , λk∼wk }
to be a (set theoretical) partition of E as a disjoint union of L sets E =
E1 ∪ E2 ∪ · · · EL . Using this definition, we have to allow that, at different
vertices, the groupings may be of different types. That is, #(Ei ) may be
different for different vertices. (One can compare with Ti −Ω < #(Ei ) < Ti +Ω
in 1.5.13.)
2. Define a grouping of
E = {λ1∼w1 , λ2∼w2 , · · · , λk∼wk }
to be a collection of L subsets E1 , E2 , · · · , EL with
∼pj1
Ej = {λ1
∼pj2
, λ2
∼pjk
, · · · , λk
},
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
297
where 0 ≤ pji ≤ wj , such that
L
X
pji = wi ,
j=1
for each i = 1, 2, · · · , k.
The grouping is called to be of type (T1 , T2 , · · · , TL ) if
#(Ej ) =
k
X
i=1
pji = Tj ,
for each j = 1, 2, · · · , L.
In this way, as will be seen, all the groupings corresponding to vertices of
a simplicial complex (as in the proof of Theorem 4.1), may be chosen to
be of the same type.
T But the conclusion (2) in the lemma should be modified. Instead of #( x∈Vertex(∆1 ) Ej (x)), j = 1, 2, · · · , L to be big, we require
◦
T
#( x∈Vertex(∆1 ) Ej (x)), j = 1, 2, · · · , L, to be big, where for any set F with
◦
multiplicity, F is the subset of F consisting of all such elements λi that {λi∼wi }
◦
are entirely inside F . (See 3.22 for detailed definition of F .)
In fact, either approaches can be carried out for our purpose. It turns out that
the second approach is shorter and more elegant. Therefore we shall take this
approach.
Even though, in our approach, it is allowed to separate some set {λi∼wi } into
different groups Ei of the grouping, we should still group as many whole sets
∼w
{λj j } of the index set {λ1∼w1 , λ2∼w2 , · · · , λn∼wn } as possible into the same set
Ei of {E1 , E2 , · · · , EL }. Assumption 3.27 and Lemma 3.28 are for this purpose.
(One needs to pay special attention to the definition and properties of GI in
3.25.) Except this idea, all other parts of the proof are the same as the case of
multiplicity one.
1.5.24. Once we have the combinatorial results in §3, and the explanations
in 1.5.1–1.5.19, it will not be hard to understand the proof of Theorem 4.1,
though there are some other small techniques, which will be clearly explained
in the proof (see 4.2–4.19).
1.5.25. Combining Theorem 4.1 and the result of [Li2] (see the lemma stated
in 1.5.11), we can obtain a decomposition φ1 ⊕ψ of φm,m′ (for m′ large enough)
such that the major part ψ factors through an interval algebra.
But to deal with the part φ1 , we should add to it, a relatively large (comparing
with φ1 ) homomorphism φ2 , which factors through a finite dimensional C ∗ algebra—or which is defined by certain point evaluations (on a δ-dense subset
of Xn,i for some small number δ).
In [Li3], Li deals with this problem by another decomposition, taking such a
homomorphism out of the part ψ. (She only proved the one dimensional case.)
We take a different approach. Going back to the construction of the maps αi
in [Li2] (see the lemma inside 1.5.11 above), we can choose sufficiently many
Documenta Mathematica 7 (2002) 255–461
298
Guihua Gong
of them to be constant maps (see Lemma 4.33 in §4 below). Therefore ψ
automatically has such a part defined by point evaluations.
We believe that our approach is easier to understand than Li’s approach, though
the spirit is the same. Furthermore, our decomposition is a quantitative version
(see Theorem 4.35), and is stronger than Li’s theorem even in the case of one
dimensional spaces. (This will be important in [EGL].)
We shall not use any result from [Li3]. But we encourage the reader to read
the short article [Li2], on which our proof heavily depends.
1.6
Some uniqueness theorems and a factorization theorem
First, this subsection contains some uniqueness theorems. In general, a uniqueness theorem states that, under certain conditions, two maps φ, ψ : A → B (homomorphisms or completely positive linear contractions between C ∗ -algebras
A and B) are approximately unitarily equivalent to each other to within a given
small number ε on a given finite set F ⊂ A, that is, there is a unitary u ∈ B
such that
kφ(f ) − uψ(f )u∗ k < ε, ∀f ∈ F.
This subsection also contains a factorization theorem, which says that, there
is a homomorphism (in the class of the so called unital simple embeddings)
between matrix algebras over (perhaps higher dimensional) spaces, which must
approximately factor through a sum of matrix algebras over the special spaces
∞
2
{pt}, [0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S , by means of almost multiplicative maps.
We put these two kinds of results together into one subsection, since the proofs
of them have some similarity. Also, in the proof of the factorization theorem,
we use some uniqueness theorems of this same subsection.
Most of the results are modifications of some results in the literature [EG2],
[D1-2], [G4] and [DG] (see [Phi], [GL], [Lin1-2] and [EGLP] also). One of the
main theorems (Theorem 1.6.9) is a generalization of Theorem 2.29 of [EG2]—
the main uniqueness theorem in the classification of real rank zero AH algebras.
The proof given here is shorter than the proof given in [EG2]. Another main
theorem—Theorem 1.6.26 (see also Corollary 1.6.29) is a refinement of Lemma
2.2 of [D2] (see also Lemma 3.13 and 3.14 of [G4]). Both Theorem 1.6.9 and
Corollary 1.6.29 are important in the proof of our main results in §6.
The following well known result (see [Lo]) will be used frequently.
Lt
Lemma 1.6.1. Suppose that A = i=1 Mki (C(Xi )), where Xi = {pt}, [0, 1], or
S 1 . For any finite set F ⊂ A and any number ε > 0, there is a finite set G ⊂ A
and there is a number δ > 0 such that if C is a C ∗ -algebra and φ ∈ Map(A, C)
is a G-δ multiplicative map, then there is a homomorphism φ′ ∈ Hom(A, C)
satisfying
kφ(f ) − φ′ (f )k < ε, ∀f ∈ F.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
299
The following result is essentially contained in [EG2] (see [EG2, 2.11]) and is
stated as Theorem 1.2 of [D1] (see also [G4, 3.2 and 3.8]).
Lemma 1.6.2.
([D1, 1.2]) Let X be a finite simplicial complex. For any
finite subset F ⊂ C(X) and any number ε > 0, there are a positive integer L,
a unital homomorphism τ : C(X) → ML (C(X)), and a unital homomorphism
µ : C(X) → ML+1 (C(X)) with finite dimensional image such that
kdiag(f, τ (f )) − µ(f )k < ε, ∀f ∈ F.
By the argument in 1.2.19, in the above lemma, the algebra C(X) can be
replaced by Mn (C(X)).
Lemma 1.6.3. Let X be a finite simplicial complex and A = Mn (C(X)). For
any finite subset F ⊂ A and any number ε > 0, there are a positive integer
L, a unital homomorphism τ : A → ML (A)(= MnL (C(X))), and a unital
homomorphism µ : A → ML+1 (A) with finite dimensional image such that
kdiag(f, τ (f )) − µ(f )k < ε, ∀f ∈ F.
Remark 1.6.4. In general, a unital homomorphism λ : C(X) → B with finite
dimensional image is always of the form:
X
λ(f ) =
f (xi )pi , ∀f ∈ F,
where {xi } is a finite
P subset of X, and {pi } ⊂ B is a set of mutually orthogonal
projections with
pi = 1B . A homomorphism λ : A = Mn (C(X)) → B with
finite dimensional image is of the form
X
λ(f ) =
pi ⊗ f (xi ), ∀f ∈ Mn (C(X))
| ), where
for a certain identification of λ(1A )Bλ(1A ) ∼
= (λ(e11 )Bλ(e11 )) ⊗ Mn (C
{pi } is a set of mutually orthogonal projections in λ(e11 )Bλ(e11 ).
The following Lemma is essentially proved in [D1, Lemma 1.4] (see [G4, Theorem 3.9] also), using the idea from [Phi] and [GL].
Lemma 1.6.5. Let X be a finite simplicial complex and A = Mn (C(X)). For
a finite set F ⊂ Mn (C(X)), a positive number ε > 0 and a positive integer N ,
there are a finite set G ⊂ Mn (C(X)), a positive number δ > 0 and a positive
integer L, such that the following is true.
For any unital C ∗ -algebra B, any N + 1 completely positive G-δ multiplicative linear ∗-contraction φ0 , φ1 , · · · φN ∈ M apG−δ (A, B), there are a homomorphism λ ∈ Hom(A, ML (B)) with finite dimensional image and a unitary
u ∈ ML+1 (B) such that
kdiag(φ0 (f ), λ(f )) − udiag(φN (f ), λ(f ))u∗ k < ε + ω, ∀f ∈ F,
Documenta Mathematica 7 (2002) 255–461
300
Guihua Gong
where
ω = max
max
f ∈F 0≤j≤N −1
kφj (f ) − φj+1 (f )k.
Proof: If we allow the number L to depend on the maps {φj }, then this is
Lemma 1.4 of [D1]. In fact, in the proof of [D1, Lemma 1.4], the author proves
this stronger version of the lemma. We will not repeat the entire proof in [D1],
instead, we will only repeat the construction of G, δ in [D1] and at the same
time choose the number L.
Apply Lemma 1.6.3 to F ⊂ A, 4ε > 0 to obtain the integer L1 , τ : A → ML1 (A)
and µ : A → ML1 +1 (A) as in Lemma 1.6.3. Then D := µ(A) is a finite
dimensional C ∗ -subalgebra of ML1 +1 (A). By Lemma 1.6.1, there are a finite
set F1 ⊂ D(⊂ ML1 +1 (A)) and a positive number δ1 > 0 such that if C is any
C ∗ -algebra and ψ ∈ Map(D, C) is any F1 -δ1 multiplicative map, then there is
a homomorphism ψ ′ ∈ Hom(D, C) such that
ε
kψ ′ (f ) − ψ(f )k < , ∀f ∈ µ(F ) (⊂ D).
4
From 1.2.19, there exist a finite set G ⊂ A and a positive number δ >
0 such that if φ ∈ Map(A, B) is G-δ multiplicative, then φ ⊗ idL1 +1 ∈
Map(ML1 +1 (A), ML1 +1 (B)) is F1 -δ1 multiplicative.
Let L := N (L1 + 1). The proof of [D1, Lemma 1.4] proves that such G, δ and
L are as desired. (Notice that the size of the homomorphism η on line 9 of
page 122 of [D1] is the number L above.)
⊔
⊓
In Lemma 1.6.5, if we further assume that F ⊂ A is weakly approximately constant to within ε, then one can replace the homomorphism λ in Lemma 1.6.5 by
an arbitrary homomorphism with finite dimensional image of sufficiently large
size (with ε replaced by 5ε). One can even use two different homomorphisms
(provided that the images of the matrix unit e11 under these two different homomorphisms are unitarily equivalent) for φ0 and φN , i.e., instead of diag(φ0 , λ)
and diag(φN , λ), one can use diag(φ0 , λ1 ) and diag(φN , λ2 ) in the estimation.
Namely, we can prove the following result.
Corollary 1.6.6. Let X be a finite simplicial complex and A = Mn (C(X)).
Suppose that ε > 0 and that a finite set F ⊂ Mn (C(X)) is weakly approximately
constant to within ε. Suppose that N is a positive integer. Then there are a
finite set G ⊂ Mn (C(X)), a positive number δ > 0, and a positive integer L
such that the following is true.
For any unital C ∗ -algebra B and projection p ∈ B, any N +1 completely positive
G-δ multiplicative linear ∗-contractions φ0 , φ1 , · · · , φN ∈ M apG−δ (A, pBp), any
λ1 , λ2 ∈ Hom(A, (1 − p)B(1 − p)) with finite dimensional images and with
λ1 (e11 ) ∼ λ2 (e11 ) (see 1.1.7(i)) and [λ1 (e11 )] ≥ L · [p], there is a unitary u ∈ B
such that
kdiag(φ0 (f ), λ1 (f )) − udiag(φN (f ), λ2 (f ))u∗ k < 5ε + ω, ∀f ∈ F,
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
301
where
ω = max
max
f ∈F 0≤j≤N −1
kφj (f ) − φj+1 (f )k.
Proof: Suppose that L and λ : A → ML (pBp) are as in Lemma 1.6.5 for the
G-δ multiplicative maps φ0 , φ1 , · · · φN ∈ Map(A, pBp).
From 1.6.4, λ is of the following form
λ(f ) =
s
X
i=1
pi ⊗ f (xi ), ∀f ∈ Mn (C(X))
| ), where
)) ⊗ Mn (C
for a certain identification of λ(1A )Bλ(1A ) ∼
= (λ(e11 )Bλ(e11P
s
pi , i = 1, 2, · · · , s, are mutually orthogonal projections with i=1 pi = λ(e11 ) ∈
ML (pBp), and {xi } ⊂ X.
Fix a base point x0 ∈ X. Since F is weakly approximately constant to within
ε, for each i, {f (xi )}f ∈F is approximately unitarily equivalent to {f (x0 )}f ∈F
to within ε, one by one by the same unitary. I.e., for each i, there is a unitary
| ) such that kvf (xi )v ∗ − f (x0 )k < ε, for any f ∈ F .
v ∈ Mn (C
Ps
Define
newλ by newλ(f ) = i=1 pi ⊗ f (x0 ) = E ⊗ f (x0 ), where E := λ(e11 ) =
Ps
i=1 pi . Then newλ is approximately unitarily equivalent to the old λ to
within ε on F . Therefore, with this newλ, we still have
(1)
kdiag(φ0 (f ), λ(f )) − u1 diag(φN (f ), λ(f ))u1 k < 3ε + ω, ∀f ∈ F,
for some unitary u1 ∈ ML+1 (pBp).
Since λ1 (e11 ) ∼ λ2 (e11 ), without loss of generality, we can assume that
2
|
λ1 |Mn (C
| ) = λ |Mn (C
| ) , where Mn (C ) ⊂ Mn (C(X)) (= A). In particular,
1
2
1
2
λ (1A ) = λ (1A ) and λ (e11 ) = λ (e11 ). Denote λ1 (e11 ) by E ′ . Similar to
the case of λ, we can assume that
λ1 (f ) =
s1
X
qi1 ⊗ f (x1i ), ∀f ∈ Mn (C(X)),
s2
X
qi2 ⊗ f (x2i ), ∀f ∈ Mn (C(X))
i=1
λ2 (f ) =
i=1
∼ (E ′ BE ′ ) ⊗ Mn (C
|
for a certain identification of λ1 (1A )Bλ1 (1A ) =
P),s1 where
1
2
1
{q
}
and
{q
}
are
two
sets
of
mutually
orthogonal
projections
with
i
i=1 qi =
Pis2 2
′
1
2
i=1 qi = E ∈ (1 − p)B(1 − p), and {xi }, {xi } ⊂ X.
Define λ̃ : A → λ1 (1A )Bλ1 (1A ) by
λ̃(f ) = E ′ ⊗ f (x0 ), ∀f ∈ F.
Similar to the argument for λ, both λ1 and λ2 are approximately unitarily
equivalent to λ̃ to within ε on F .
Documenta Mathematica 7 (2002) 255–461
302
Guihua Gong
Since [E] ≤ L·[p] ≤ [E ′ ] (= [λ1 (e11 )]), there is a sub-projection E1 ≤ E ′ which
is unitarily equivalent to E.
Write λ̃ = µ1 ⊕ µ2 , where µ1 (f ) = E1 ⊗ f (x0 ) and µ2 (f ) = (E ′ − E1 ) ⊗ f (x0 ).
Then µ1 is unitarily equivalent to λ (strictly speaking, newλ). From (1), we
have
kdiag(φ0 (f ), µ1 (f )) − u2 diag(φN (f ), µ1 (f ))u∗2 k < 3ε + ω, ∀f ∈ F
for a unitary u2 ∈ (p⊕E1 ⊗1n )B(p⊕E1 ⊗1n ). Notice that E1 ⊗1n ≤ E ′ ⊗1n =
λ1 (1A ) ≤ (1 − p).
Therefore,
kdiag(φ0 (f ), λ̃(f )) − u3 diag(φN (f ), λ̃(f ))u∗3 k < 3ε + ω, ∀f ∈ F,
where u3 := u2 ⊕ ((E ′ − E1 ) ⊗ 1n ) ∈ (p ⊕ (E ′ ⊗ 1n ))B(p ⊕ (E ′ ⊗ 1n )).
We already know that both λ1 and λ2 are approximately unitarily equivalent
to λ̃ on F to within ε, so we have
kdiag(φ0 (f ), λ1 (f )) − udiag(φN (f ), λ2 (f ))u∗ k < 5ε + ω, ∀f ∈ F,
for a unitary u ∈ B.
⊔
⊓
Lemma 1.6.7. Suppose that A = Mk (C(X)), and F ⊂ A is weakly approximately constant to within ε. Suppose that A1 is a C ∗ -algebra, and two homomorphisms φ and ψ ∈ Hom(A, A1 ) are homotopic to each other. There are a
finite set G ⊂ A1 , a number δ > 0, and a positive integer L > 0 such that the
following is true.
If B is a unital C ∗ -algebra, p ∈ B is a projection, λ0 ∈ M ap(A1 , pBp) is G-δ
multiplicative, λ1 ∈ Hom(A1 , (1 − p)B(1 − p)) is a homomorphism with finite
dimensional image satisfying [(λ1 ◦ φ)(e11 )] ≥ L · [p], and λ ∈ M ap(A1 , B) is
defined by λ = λ0 ⊕ λ1 , then there is a unitary u ∈ B such that
k(λ ◦ φ)(f ) − u(λ ◦ ψ)(f )u∗ k < 6ε, ∀f ∈ F.
Proof: Since φ is homotopic to ψ. There is a continuous path of homomorphisms φt , 0 ≤ t ≤ 1, such that φ0 = φ and φ1 = ψ. Choose
0 = t0 < t1 < · · · tN −1 < tN = 1 such that
kφtj+1 (f ) − φtj (f )k < ε, ∀j ∈ {0, 1, · · · , N − 1} and ∀f ∈ F.
Applying Corollary 1.6.6 to ε, F ⊂ A (which is weakly approximately constant
to within ε), and the number N from the above, there are G1 ⊂ A and δ > 0
and L as in the
SNCorollary 1.6.6.
The set G := j=0 φtj (G1 ) ⊂ A1 , δ > 0 and number L are as desired.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
303
Suppose that λ0 , λ1 are the maps satisfying the conditions described in the
lemma for G, δ, and L as chosen above. Choosing the sequence of G1 -δ multiplicative maps in Corollary 1.6.6 to be λ0 ◦φt0 (= λ0 ◦φ), λ0 ◦φt1 , · · · , λ0 ◦φtN (=
λ0 ◦ ψ), and the homomorphisms λ1 and λ2 (with finite dimensional images)
to be λ1 = λ1 ◦ φ and λ2 = λ1 ◦ ψ, and using that λ = λ0 ⊕ λ1 , we have
k(λ ◦ φ)(f ) − u(λ ◦ ψ)(f )u∗ k < 5ε + ω, ∀f ∈ F,
for a certain unitary u ∈ B, where
ω = max
max
f ∈F 0≤j≤N −1
k(λ0 ◦ φtj+1 )(f ) − (λ0 ◦ φtj )(f )k < ε,
since λ0 is a contraction—norm decreasing map. So the Lemma follows. (Note
that if λ0 is G-δ multiplicative, then λ0 ◦ φtj is G1 -δ multiplicative. Also note
that we have the condition that [λ1 ◦ φ(e11 )] ≥ L · [p]. Another condition
λ1 ◦ φ(e11 ) ∼ λ1 ◦ ψ(e11 ) follows from the condition φ ∼h ψ.)
⊔
⊓
The author is indebted to Professor G. Elliott for pointing out the proof of the
following result to him.
Lemma 1.6.8. Suppose that C is a unital C ∗ -algebra, and D ⊂ C is a finite
dimensional C ∗ -subalgebra. For any finite set F ⊂ C and any positive number
ε > 0, there are a finite set G ⊂ C and a number δ > 0 such that if B
is a unital C ∗ -algebra, λ ∈ M ap(C, B) is G-δ multiplicative, then there is a
λ′ ∈ M ap(C, B) satisfying the following conditions.
1. λ′ |D is a homomorphism.
2. kλ′ (f ) − λ(f )k < ε, ∀f ∈ F .
Proof: Without loss of generality, we assume that kf k ≤ 1 for all f ∈ F .
By Kasparov’s version of Stinespring Dilation Theorem, for the completely
positive linear ∗-contraction λ : C → B, there is a homomorphism φ : C →
M (B ⊗ K) such that λ(f ) = pφ(f )p for all f ∈ F , where K is the algebra of all
compact operators on an infinite dimensional separable Hilbert space, M (B⊗K)
is the multiplier algebra of B ⊗ K, and p = 1B ⊗ e11 ∈ B ⊗ K ⊂ M (B ⊗ K).
For the above λ and φ, it is straight forward to check that, for any fixed element
a ∈ C, if kλ(a · a∗ ) − λ(a)λ(a∗ )k < δ, then kpφ(a)(1 − p) · (pφ(a)(1 − p))∗ k < δ.
Therefore, if we choose the finite set G to satisfy that G = G∗ , then the G-δmultiplicativity of the map λ implies the following property of the dilation φ
and the cutting down projection p:
√
(∗)
kφ(a) − (pφ(a)p + (1 − p)φ(a)(1 − p))k < 2 δ, ∀a ∈ G,
where 1 is the unit of M (B ⊗ K).
By a well known perturbation technique (see [Gli] and [Br]), we have the following: If G contains all matrix units eij of each block of D and δ is small enough,
Documenta Mathematica 7 (2002) 255–461
304
Guihua Gong
then the above condition (*) implies that there is a unitary u ∈ M (B ⊗ K) with
ku − 1k < 2ε , such that
uφ(D)u∗ ⊂ pM (B ⊗ K)p ⊕ (1 − p)M (B ⊗ K)(1 − p).
(One can obtain the above assertion by applying Lemma III.3.2 of [Da] (or even
stronger result of [Ch]) with φ(D) and pM (B ⊗ K)p ⊕ (1 − p)M (B ⊗ K)(1 − p)
in place of U and B in [Da, III.3.2], respectively.)
The map λ′ : C → B, defined by λ′ (f ) = puφ(f )u∗ p, is as desired.
⊔
⊓
The following result can be considered to be a generalization of Theorem 2.29
of [EG 2].
Ls
Theorem 1.6.9. Suppose that A =
i=1 Mki (C(Xi )) and F ⊂ A is weakly
approximately constant to within ε. Suppose that C is a C ∗ -algebra, the homomorphisms φ and ψ ∈ Hom(A, C) are homotopic to each other. There are a
finite set G ⊂ C, a number δ > 0, and a positive integer L > 0 such that the
following is true.
If B is a unital C ∗ -algebra, p ∈ B is a projection, λ0 ∈ M ap(C, pBp) is G-δ
multiplicative, λ1 ∈ Hom(C, (1 − p)B(1 − p)) is a homomorphism with finite
dimensional image satisfying that for each i ∈ {1, 2, · · · , s}, [(λ1 ◦ φ)(ei11 )] ≥
L · [p], where ei11 is the matrix unit (of upper left corner ) of the i-th block,
Mki (C(Xi )), of A, then there is a unitary u ∈ B such that
k(λ ◦ φ)(f ) − u(λ ◦ ψ)(f )u∗ k < 8ε, ∀f ∈ F,
where λ ∈ M ap(A1 , B) is defined by λ = λ0 ⊕ λ1 .
Proof: Let φt be the homotopy between φ and ψ. It is well known that there
is a unitary path ut ∈ C such that
φt (1Ai ) = ut φ0 (1Ai )u∗t ,
for all blocks Ai = Mki (C(Xi )). Therefore, without loss of generality, we assume that φ(1Ai ) = ψ(1Ai ), and that φ|Ai is homotopic to ψ|Ai within the
corner φ(1Ai )Cφ(1Ai ).
Apply Lemma 1.6.7 to φ|Ai , ψ|Ai and πi (F ), where πi is the quotient map from
A to Ai , to obtain G1 (⊂ C), δ1 and L as G, δ and L in Lemma 1.6.7. For
convenience, without loss of generality, we assume that kgk ≤ 1 for all g ∈ G 1 .
Consider the finite dimensional subalgebra
Let Ei = φ(1Ai ).
| · E1 ⊕ C
| · E2 ⊕ · · · ⊕ C
| · Es ⊂ C. Applying Lemma 1.6.8, there
D := C
are G ⊂ C with G ⊃ G1 and δ > 0 with δ < δ31 such that if λ0 ∈ Map(C, pBp)
is G-δ multiplicative, then there is another map λ′0 ∈ Map(C, pBp) satisfying
the following conditions.
1. The restriction λ′0 |D is a homomorphism.
2. kλ′0 (f ) − λ0 (f )k < min( δ31 , ε), ∀f ∈ G1 ∪ φ(F ) ∪ ψ(F ).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
305
As a consequence we also have
3. λ′0 is G1 -δ1 multiplicative.
The condition 1 above yields that {λ′0 (Ei )}si=1 are mutually orthogonal projections.
Such G, δ and L are as desired.
Suppose that λ0 and λ′0 are as above. Set λ′ = λ′0 ⊕ λ1 . From Lemma 1.6.7
and the ways G1 , δ1 and L are chosen, there are unitaries ui ∈ λ′ (Ei )Bλ′ (Ei )
such that
k(λ′ ◦ φ|Ai )(f ) − ui (λ′ ◦ ψ|Ai )(f )u∗i k < 6ε, ∀f ∈ Fi .
L
P
Then the unitary u = i ui ⊕ (1 − i λ′ (Ei )) satisfies
k(λ′ ◦ φ)(f ) − u(λ′ ◦ ψ)(f )u∗ k < 6ε, ∀f ∈ F.
Hence
k(λ ◦ φ)(f ) − u(λ ◦ ψ)(f )u∗ k < 6ε + 2ε = 8ε, ∀f ∈ F.
⊔
⊓
Remark 1.6.10. The version of Theorem 2.29 of [EG2] with A being a direct
sum of full matrix algebras is a direct consequence of the above theorem and
Corollary 2.24 of [EG2] (see [EG2, Theorem 2.21] also). In order to obtain
the general version of Theorem 2.29 of [EG2], one needs to apply the dilation
lemma [EG2, 2.13] and Lemma 1.6.8 above. (The number 8ε should be changed
to 5 · 8ε = 40ε which is still better than 70ε in [EG2].)
The following lemma is a direct consequence of Lemma 1.6.5.
Lemma 1.6.11. Let X be a finite simplicial complex and A = C(X). Let
F ⊂ A be a finite set and ε > 0. There are a finite set G ⊂ A and a number
δ > 0 with the following property.
If B is a unital C ∗ -algebra, φt : A → B, 0 ≤ t ≤ 1 is a continuous path of
G-δ multiplicative maps (i.e., φt ∈ MapG−δ (A, B)), then there are a positive
integer L, a homomorphism λ : A → ML (B) with finite dimensional image,
and a unitary u ∈ ML+1 (B) such that
k(φ0 ⊕ λ)(f ) − u(φ1 ⊕ λ)(f )u∗ k < ε, ∀f ∈ F.
The proof of the following corollary has some similarity to the proof of Corollary
1.6.6. Such method will be used frequently.
Corollary 1.6.12. Let X be a finite simplicial complex and A = C(X). Let
F ⊂ A be a finite set and ε > 0. There are a finite set G ⊂ A and a number
δ > 0 with the following property.
If B is a unital C ∗ -algebra, p ∈ B is a projection, φt : A → pBp, 0 ≤ t ≤ 1 is a
continuous path of G-δ multiplicative maps , then there are a positive integer L
Documenta Mathematica 7 (2002) 255–461
306
Guihua Gong
and a number η > 0 such that for any η-dense subset {x1 , x2 , · · · , x• } ⊂ X, any
set of mutually orthogonal projections {p1 , p2 , · · · , p• } ⊂ B ⊗K with [pi ] ≥ L·[p]
and pi ⊥p, we have
kφ0 (f ) ⊕
•
X
i=1
f (xi )pi − u(φ1 (f ) ⊕
•
X
i=1
f (xi )pi )u∗ k < ε, ∀f ∈ F,
for a certain unitary u ∈ (p ⊕ p1 ⊕ · · · p• )(B ⊗ K)(p ⊕ p1 ⊕ · · · p• ).
Proof: For the finite set F ⊂ A, choose η small enough such that if dist(x, x′ ) <
η, then kf (x) − f (x′ )k < 3ε for all f ∈ F . Apply Lemma 1.6.11 to F and 3ε
to obtain G and δ. For the path φt : A → pBp, there exist a positive integer
L, a homomorphism λ : A → ML (pBp), and a unitary u1 ∈ ML+1 (pBp) as in
Lemma 1.6.11. That is
ε
k(φ0 ⊕ λ)(f ) − u1 (φ1 ⊕ λ)(f )u∗1 k < , ∀f ∈ F.
3
From 1.6.4, λ is of the form
λ(f ) =
l
X
f (yi )qi ,
i=1
where {y1 , y2 , · · · , yl } ⊂ X, and {q1 , q2 , · · · , ql } ⊂ ML (pBp) is a set of mutually
orthogonal projections.
Since {x1 , x2 , · · · , x• } is an η-dense subset of X, we can divide the set
{y1 , y2 , · · · , yl } into a disjoint union of subsets X1 ∪ X2 ∪ · · · ∪ X•P
(some Xi may
be empty) such that dist(y, xi ) < η for any y ∈ Xi . Set p′i := yj ∈Xi qj and
P•
define λ′ : C(X) → ML (pBp) by λ′ (f ) = i=1 f (xi )p′i . (Note that for some i,
p′i might be 0.) Then from the way η is chosen, we have
kλ′ (f ) − λ(f )k <
Therefore,
ε
, ∀f ∈ F.
3
k(φ0 ⊕ λ′ )(f ) − u1 (φ1 ⊕ λ′ )(f )u∗1 k < ε, ∀f ∈ F.
Our corollary follows from the fact [p′i ] ≤ L · [p] ≤ pi .
⊔
⊓
If X does not contain any isolated point, then in the above corollary, we can
change the condition [pi ] ≥ L·[p] to a weaker condition [pi ] ≥ [p], by choosing η
smaller. (Roughly speaking, this is true because η could be chosen so small that
if {xi } is η-dense, then there are at least L points of xi in the η ′ -neighborhood
of any point in X for a pre-given small number η ′ . If X is a space of single
point, then this is not true.) Therefore, the number L does not appear in the
following corollary.
Corollary 1.6.13. Let X be a finite simplicial complex without any single
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
307
point components, A = C(X). Let F ⊂ A be a finite set and ε > 0. There are
a finite set G ⊂ A and a number δ > 0 with the following property.
If B is a unital C ∗ -algebra, p ∈ B is a projection, φt : A → pBp, 0 ≤ t ≤ 1,
is a continuous path of G-δ multiplicative maps , then there is a number η > 0
such that for any η-dense subset {x1 , x2 , · · · , x• } ⊂ X, any set of mutually
orthogonal projections {p1 , p2 , · · · , p• } ⊂ B ⊗ K with [pi ] ≥ [p] and pi ⊥p, we
have
kφ0 (f ) ⊕
•
X
i=1
f (xi )pi − u(φ1 (f ) ⊕
•
X
i=1
f (xi )pi )u∗ k < ε, ∀f ∈ F
for a certain unitary u ∈ (p ⊕ p1 ⊕ · · · p• )(B ⊗ K)(p ⊕ p1 ⊕ · · · p• ).
Proof: Let L and η1 (in place of η) be as in Corollary 1.6.12. Let η2 be the
minimum of the diameters of path connected components of X, which is positive
since X has no single point component. And let η3 be a positive number such
that if dist(x, x′ ) < η3 , then kf (x) − f (x′ )k < ε.
η′
.
Define η ′ = min(η1 , η2 , η3 ). Let η = 8L
Suppose that X ′ = {x1 , x2 , · · · , x• } is an η-dense finite subset of X. Choose a
′
η ′ -dense subset {xk1 , xk2 , · · · , xkl } ⊂ X ′ such that dist(xki , xkj ) ≥ η2 if i 6= j.
(Such subset exists. It could be chosen to be a maximum subset of X ′ such
′
that the distance of any two points in the set is at least η2 . Then the η ′ -density
follows from the maximality.) It is easy to see that there is a partition of X ′
as X ′ = X1 ∪ X2 ∪ · · · ∪ Xl such that
X ′ ∩ B η′ (xki ) ⊂ Xi ⊂ X ′ ∩ Bη′ (xki ).
4
Since X ′ is η-dense and η =
η′
8L ,
#(Xi ) ≥ #(X ′ ∩ B η′ (xki )) ≥ L.
4
(Here we also use the fact that the connected component of xki in X has
diameter at least η ′ . )
Let pj , j = 1, 2, · · · •, be the projections as in the corollary. Define qi =
P
xj ∈Xi pj , i = 1, 2, · · · l. Then from [pj ] ≥ [p] and #(Xi ) ≥ L, we have,
[qi ] ≥ L · [p].
Our corollary (with 3ε in place of ε) follows from an application of Corollary
1.6.12 to {xki }li=1 and {qi }li=1 , and the estimation
k
•
X
i=1
f (xi )pi −
l
X
i=1
f (xki )qi k < ε, ∀f ∈ F.
(The above estimation is a consequence of the way η3 is chosen and the fact
that Xi ⊂ Bη′ (xki ) with η ′ < η3 .)
⊔
⊓
Documenta Mathematica 7 (2002) 255–461
308
Guihua Gong
The following lemma is proved by applying Lemma 1.6.8.
L k
Ll
Lemma 1.6.14. Let A =
A =
k=1 Ms(k) (C(Xk )), where Xk are connected simplicial complexes and {s(k)} are positive integers. For any finite
set G′ ⊂ A, any number δ ′ > 0, any finite sets Gk1 ⊂ C(Xk ) and any numbers
δ1k > 0, k = 1, 2, · · · l, there are a finite set G ⊂ A and a number δ > 0 such that
if φ ∈ Map(A, B) is G-δ multiplicative, then there is a map φ′ ∈ Map(A, B)
satisfying the following conditions.
(1) φ′ is G′ -δ ′ multiplicative;
(2) kφ′ (g) − φ(g)k < δ ′ for all g ∈ G;
(3) {φ′ (1Ak )}lk=1 are mutually orthogonal projections in B and φ′ (ek11 ) ∈ B
are subprojections of φ′ (1Ak ) ∈ B. And if each φk1 ∈ Map(C(Xk ), B) is
the restriction of φ′ on ek11 Ms(k) (C(Xk ))ek11 ∼
= C(Xk ), then one can identify
φ′ (1Ak )Bφ′ (1Ak ) with φ′ (ek11 )Bφ′ (ek11 ) ⊗ Ms(k) such that
φ′ =
l
M
k=1
Furthermore,
φk1
is
Gk1 -δ1k
φk1 ⊗ ids(k) .
Multiplicative.
Proof: The part of Gk1 -δ1k multiplicativity of φk1 follows from the G′ -δ ′ multiplicativity of φ′ if we enlarge the set G′ and reduce the number δ ′ so that
G′ ⊃ {g · ek11 | g ∈ Gk1 } and δ ′ < δ1k . Also we can assume that G′ contains
{ekij }—the set of all matrix units.
By Lemma 1.6.8, without loss of generality, we assume that the restriction
φ|Ll M (C
| ) is a homomorphism.
s(k)
k=1
Ll
Let φk1 = φ|ek11 Aek11 ∈ Map(C(Xk ), φ(ek11 )Bφ(ek11 )). Then φ′ := k=1 φk1 ⊗ids(k)
is defined by
X
φ′ (f ) =
φ(eki1 )φ(fij · ek11 )φ(ek1j ),
i,j
k
where f = (fij )s(k)×s(k) ∈ A .
P
k
k
Note that for the above f ∈ Ak , one can write f =
i,j ei1 · (fij · e11 ) ·
ek1j . Obviously, if we choose G to be the set of all the elements which can be
expressed as products of at most ten elements from the set G′ , and if we choose
δ small enough, then (1) and (2) will hold for φ′ . (Notice that G′ contains all
the matrix units.)
⊔
⊓
The following result follows from Lemma 1.6.14 and Corollary 1.6.12 (see also
1.2.19).
Ll
Corollary 1.6.15. Let A =
k=1 Ms(k) (C(Xk )), where Xk are connected
finite simplicial complexes and s(k) are positive integers. Let F ⊂ A be a finite
set and ε > 0. There are a finite set G ⊂ A and a number δ > 0 with the
following property.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
309
If B is a unital C ∗ -algebra, p ∈ B is a projection, φt : A → pBp, 0 ≤ t ≤ 1 is
a continuous path of G-δ multiplicative maps , then there are a positive integer
L, and η > 0 such that for a homomorphism λ : A → B ⊗ K (with finite
dimensional image), there is a unitary u ∈ B satisfying:
kφ0 (f ) ⊕ λ(f ) − u(φ1 (f ) ⊕ λ(f ))u∗ k < ε, ∀f ∈ F,
provided that λ is`of the following form: there are an η-dense subset
l
{x1 , x2 , · · · , x• } ⊂ k=1 Xk (=
set of mutually orthogonal
LSP(A)), and a L
projections {p1 , p2 , · · · , p• } ⊂ λ( k ek11 )(B ⊗ K)λ( k ek11 ) with [pi ] ≥ L · [p],
such that
•
X
pi ⊗ f (xi ), ∀f ∈ A
λ(f ) =
i=1
| ).
under the identification λ(1Ak )Bλ(1Ak ) ∼
= (λ(ek11 )Bλ(ek11 )) ⊗ Ms(k) (C
Proof: One can apply Lemma 1.6.14 to φt ∈ Map(A, pBp[0, 1]) to reduce the
problem to the case of A = C(Xk ) which is Corollary 1.6.12. (Here pBp[0, 1]
is defined to be the C ∗ -algebra of continuous pBp valued functions on [0, 1].)
⊔
⊓
For convenience, we introduce the following definitions.
Ln
Definition 1.6.16. A homomorphism φ : A =
i=1 Mki (C(Xi )) → B =
Ln′
(C(Y
))
is
called
m-large
if
for
each
partial
map φij : Ai =
M
j
lj
j=1
Mki (C(Xi )) → B j = Mlj (C(Yj )) of φ,
rank(φij (1Ai ) ≥ m · rank(1Ai )(= m · ki ).
Definition 1.6.17. Let X be a connected finite simplicial complex, A =
Mk (C(X)). A unital *-monomorphism φ : A → Ml (A) is called a (unital)
simple embedding if it is homotopic to the homomorphism id ⊕ λ, where
λ : A → Ml−1 (A) is defined by
λ(f ) = diag(f (x0 ), f (x0 ), · · · , f (x0 )),
|
{z
}
l−1
for a fixed
point x0 ∈ X.
Lbase
n
Let A = i=1 Mki (C(Xi )), where Xi are connected finite simplicial complexes.
A unital *-monomorphism φ : A → Ml (A) is called a (unital) simple embedding, if φ is of the form φ = ⊕φi defined by
φ(f1 , f2 , · · · , fn ) = (φ1 (f1 ), φ2 (f2 ), · · · , φn (fn )),
where the homomorphisms φi : Ai (= Mki (C(Xi ))) → Ml (Ai ) are unital simple
embeddings.
Documenta Mathematica 7 (2002) 255–461
310
Guihua Gong
1.6.18. For each connected finite simplicial complex X, there is a three dimensional connected simplicial complex Y = Y1 ∨ Y2 ∨ · · · ∨ Y• such that
K ∗ (X) = K ∗ (Y ), where Yi are the following special spaces: [0, 1], S 1 ,
∞
2
{TII,k }∞
k=2 , {TIII,k }k=2 and S .
The space [0, 1] could be avoided in the construction of Y . But we would like to
use the space [0, 1] for the following special case: If K 0 (X) = ZZ and K 1 (X) = 0
(e.g., X is contractible) and X is not the space of a single point, then we choose
Y = [0, 1]. When X is the space of a single point, choose Y = {pt}.
The following result is Lemma 2.1 of [D2] (see Lemma 3.13 and Lemma 3.14
of [G4] also).
Ls
Lemma 1.6.19. ([D2, 2.1]) Let B1 =
j=1 Mk(j) (C(Yj )), where Yj are the
∞
2
following spaces: {pt}, [0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S . Let X be a
connected finite simplicial complex, let Y be the three dimensional space defined
in 1.6.18 with K ∗ (X) = K ∗ (Y ), and let A = MN (C(X)).
Let α1 : B1 → A be a homomorphism. For any finite sets G ⊂ B1 and F ⊂ A,
and any number δ > 0, there exists a diagram
A
x
α1
B1
φ
−→
β
ց
ψ
′
A
x
α2
−→ B2 ,
where
A′ = ML (A), B2 = MS (C(Y ));
ψ is a homomorphism, α2 is a unital homomorphism, and φ is a unital simple
embedding (see 1.6.17);
β ∈ Map(A, B2 ) is F -δ multiplicative.
Moreover there exist homotopies Ψ ∈ Map(B1 , B2 [0, 1]) and Φ ∈
Map(A, A′ [0, 1]) such that Ψ is G-δ multiplicative, Φ is F -δ multiplicative, and
Ψ|1 = ψ, Ψ|0 = β ◦ α1 , Φ|0 = α2 ◦ β and Φ|1 = φ.
(In the application of this lemma, it is important to require that φ is a unital
simple embedding. This requirement means that φ defines the same element in
kk(X, X) (connective KK theory) as the identity map id : A → A. Roughly
speaking, this lemma (and Theorem 6.26 below) means that an “identity map”
could factor through matrix algebras over Y — a special space of dimension
three.)
Proof: If one assumes that α1 : B1 → A is m-large (see 1.6.16) for a number
m > 4 dim(X), then this lemma becomes Lemma 2.1 of [D2]. We make use of
this special case to prove the general case as below.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
311
Define a unital simple embedding λ : A → Mm (A) (m > 4 dim(Y )) by
λ(f ) = diag(f, f (x0 ), f (x0 ), · · · , f (x0 )).
{z
}
|
m−1
Then α1′ = λ ◦ α1 is m-large. Apply Lemma 2.1 of [D2]— the special case of
the lemma to α1′ , λ(F ) ⊂ Mm (A) and G ⊂ B1 to obtain the following diagram
Mmx(A)
′
α1
B1
φ′
′
A
x
′
α2
−→
β′
ց
ψ′
−→
B2
with homotopy paths Ψ′ and Φ′ with properties described in the lemma for the
homomorphism α1′ , finite sets λ(F ) ⊂ Mm (A) and G ⊂ B1 .
Define β = β ′ ◦ λ, φ = φ′ ◦ λ, α2 = α2′ , ψ = ψ ′ , Ψ = Ψ′ and Φ = Φ′ ◦ λ. Then
we have the desired diagram with the desired properties.
⊔
⊓
Remark 1.6.20.
From the construction of φ and α2 in the proof of [D2,
Lemma 2.1], we know that φ and α2 take trivial projections to trivial projections. But ψ may not take trivial projections to trivial projections unless α 1
does.
1.6.21.
Let X and Y be path connected finite simplicial complexes, and
C = Mk (C(Y )), D = Ml (C(X)). Let x0 ∈ X and y0 ∈ Y be fixed base points.
Then from Lemma 3.14 of [EG2], we have the following: any homomorphism
φ ∈ Hom(C, D) is homotopy equivalent to a homomorphism φ′ ∈ Hom(C, D)
(within Hom(C, D)) such that φ′ (C 0 ) ⊂ D0 , where C 0 , D0 , are the ideals of C
and D, respectively, which consist of matrix valued functions vanishing on the
| ) such
base points (see 1.1.7(h)). In other words, there is a unitary U ∈ Ml (C
that
f (y0 )
′
φ (f )(x0 ) = U
..
.
f (y0 )
0
..
.
0
∗
U ∈ Ml (C
| ) ∀f ∈ C.
Notice that if a homomorphism α2 is as desired in Lemma 1.6.19, then any
homomorphism, which is homotopic to α2 , is also as desired. Therefore, in
Lemma 1.6.19, we can require that the homomorphism α2 : B2 (= MS (C(Y )) →
A′ (= ML (C(X))) is of the above form for certain base points y0 ∈ Y and
x0 ∈ X.
Documenta Mathematica 7 (2002) 255–461
312
Guihua Gong
In the following, let us explain why we can also choose the homomorphism α 2
to be injective.
If X is the space of a single point, then Y is also the space of a single point by
our choice. And therefore, α2 is injective, since B2 is simple.
If the connected simplicial complex X is not a single point (i.e., dim(X) ≥ 1),
then it can be proved that there is a continuous surjective map g : X → Y ,
using the standard idea of Peano curve. In fact, one can assume that the map
g is homotopy trivial—one can make it factor through an interval.
On the other hand, by Theorem 6.4.4 of [DN], if L ≥ 3S(dim(X) + 1), then
for the unital homomorphism α2 : MS (C(Y )) → ML (C(X)), there is a homomorphism α′ : MS (C(Y )) → ML−S (C(X)) such that α2 is homotopic to the
homomorphism defined by
f 7→ diag(α′ (f ), f (y0 )),
Then α2 is homotopic to diag(α′ , g ∗ ) defined by
f 7→ diag(α′ (f ), f ◦ g),
since g is homotopy trivial. So we can replace α2 by diag(α′ , g ∗ ) which is
injective, since g is surjective.
Similarly, if SP(B2 ) = Y is not a single point space (i.e., X is not the space
of a single point), then the homomorphism ψ : B1 → B2 could be chosen to
be injective with in the same homotopy class of Hom(B1 , B2 ), provided that
α1 (1B1i ) 6= 0, for each block B1i of B1 (later on, we will always assume α1
satisfies this condition, since otherwise this block can be deleted from B 1 ).
Lemma 1.6.22. Let B = Mk (C(Y )), A = Ml (C(X)). Suppose that a unital homomorphism α : B → A satisfies α(B 0 ) ⊂ A0 , and takes any trivial
projections of B to trivial projections of A, where B 0 = Mk (C0 (Y )) := {f ∈
Mk (C(Y )) | f (y0 ) = 0}, and A0 = Ml (C0 (X)) := {f ∈ Ml (C(X)) | f (x0 ) =
0}, for some fixed base points y0 ∈ Y and x0 ∈ X. Let β0 : B → Mn (B) and
β1 : A → Mn (A) be unital homomorphisms defined by
β0 (f )(y) = diag(f (y), f (y0 ), · · · , f (y0 )), ∀f ∈ B,
and
β1 (f )(x) = diag(f (x), f (x0 ), · · · , f (x0 )), ∀f ∈ A.
Then the following diagram commutes up to unitary equivalence.
Ax
α
B
β
1
−→
β
0
−→
Mxn (A)
α⊗idn
Mn (B).
I.e., there is a unitary u ∈ Mn (A) such that
β1 ◦ α = Adu ◦ (α ⊗ idn ) ◦ β0 .
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
313
Proof: β1 ◦ α is defined by:
f 7→ α(f ) 7→ diag(α(f ), α(f )(x0 ), · · · , α(f )(x0 )),
and (α ⊗ idn ) ◦ β0 is defined by:
f 7→ diag(f, f (y0 ), · · · , f (y0 )) 7→ diag(α(f ), α(f (y0 )), · · · , α(f (y0 ))),
where α(f (y0 )) denotes the result of α acting on the constant function g =
f (y0 ).
On the other hand, from α(B 0 ) ⊂ A0 , we get
α(f )(x0 ) ∼u diag(f (y0 ), · · · , f (y0 )),
|
{z
}
l
k
and from the fact that α takes trivial projections to trivial projections, we get
diag(f (y0 ), · · · , f (y0 )) ∼u α(f (y0 )),
{z
}
|
l
k
where the symbol ∼u means to be unitarily equivalent.
The following result is from [EG2] (see 5.10, 5.11 of [EG2]).
⊔
⊓
Lemma 1.6.23. Let Y = Y1 ∨Y2 ∨· · ·∨Ym . If n is large enough, then any unital
homomorphism β : Mk (C(Y )) → Mnk (C(Y )) is homotopic
to a homomorphism
Lm
β ′ : Mk (C(Y )) → Mnk (C(Y )), which factors through
i=1 Mki (C(Yi )), for
certain integers {ki }, as
β1
β ′ : Mk (C(Y )) −→
m
M
i=1
β2
Mki (C(Yi )) −→ Mnk (C(Y )).
Furthermore, β1 and β2 above can be chosen to be injective.
Proof: If k = 1, then the lemma is Lemma 5.11 of [EG2]. (Notice that, we
choose both spaces X and Y in Lemma 5.11 of [EG2] to be the above space
Y . In addition, the spaces Xi in Lemma 5.11 of [EG2] could be chosen to be
spaces Yi in our case, according to 5.10 of [EG2].)
For the general case, one writes β as b ⊗ idk , where b = β|e11 Mk (C(Y ))e11 :
C(Y ) → β(e11 )Mnk (C(Y ))β(e11 ), then apply Lemma 5.11 of [EG2] to b.
Furthermore, one can make β1 and β2 injective in the same way as in the end
of 1.6.21. (Or one observes that the maps β1 and β2 constructed in Lemma
5.11 of [EG2] are already injective for our case.)
⊔
⊓
Documenta Mathematica 7 (2002) 255–461
314
Guihua Gong
Combining Lemmas 1.6.19, 1.6.21, 1.6.22 and 1.6.23, we have the following
Lemma:
Ls
Lemma 1.6.24. Let B1 =
j=1 Mk(j) (C(Yj )), where Yj are spaces: {pt},
1
∞
∞
[0, 1], S , {TII,k }k=2 , {TIII,k }k=2 , and S 2 . Let X be a connected finite simplicial complex and let A = MN (C(X)).
Let α1 : B1 → A be a homomorphism with α1 (1B1i ) 6= 0 for each block B1i of
B1 . For any finite sets G ⊂ B1 and F ⊂ A, and any number δ > 0, there exists
a diagram
A
x
α1
B1
φ
−→
β
ց
ψ
−→
′
A
x
α2
B2 ,
where
A′ = ML (A), and B2 is a direct sum of matrix algebras over the spaces: {pt},
∞
2
[0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S ;
ψ is a homomorphism, α2 is a unital injective homomorphism, and φ is a unital
simple embedding (see 1.6.17).
β ∈ Map(A, B2 ) is F -δ multiplicative.
Moreover, there exist homotopies Ψ ∈ Map(B1 , B2 [0, 1]) and Φ ∈
Map(A, A′ [0, 1]) such that Ψ is G-δ multiplicative, Φ is F -δ multiplicative, and
Ψ|1 = ψ, Ψ|0 = β ◦ α1 , Φ|0 = α2 ◦ β and Φ|1 = φ.
Furthermore, if X is not the space of a single point, then at least one of the
blocks of B2 has spectrum different from the space of single point and ψ can be
chosen to be injective.
Proof: Let
A
x
α1
B1
φ
−→
β
ց
ψ
−→
′
A
x
α2
B2 ,
be the diagram described in Lemma 1.6.19 with homotopies Φ and Ψ. Let n
be the integer obtained by applying Lemma 1.6.23 to B2 = Mk (C(Y )). Then
apply 1.6.22 to α2 : B2 → A′ to obtain a diagram
′
Ax
α2
B2
β
1
−→
β
0
−→
′
M
xn (A )
α2 ⊗idn
Mn (B2 ).
which commutes up to homotopy. (Here we have the condition that α2 takes
trivial projections to trivial projections from Remark 1.6.20. Also, α2 is homotopic to a homomorphism which takes B 0 to A′0 .)
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
315
Furthermore, from Lemma 1.6.23, β0 is homotopic to a homomorphism β0′
factoring through a C ∗ -algebra newB2 which is a direct sum of matrix algebras
∞
2
over spaces {pt}, [0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S . Now it is routine
to finish the construction of the diagram. We omit the details.
⊔
⊓
1.6.25. Our next task is to add a homomorphism λ : A → Mn (B2 ) into the
diagram in Lemma 1.6.24 to obtain diagrams,
A
x
α1
B1
β⊕λ
ց
ψ⊕(λ◦α1 )
−→
Mn+1 (B2 )
and
A
φ⊕((α2 ⊗idn )◦λ)
−→
β⊕λ
ց
Mxn+1 (A′ )
α2 ⊗idn+1
Mn+1 (B2 )
which are almost commutative up to unitary equivalence, using Corollary
1.6.15.
To do so, we make the following assumption.
Assumption: α1 : B1 → A is injective.
Let G1 ⊂ B1 and F1 ⊂ A be any finite sets, and ε > 0. We will make the above
diagrams approximately commute on G1 and F1 , respectively, to within ε, up
to unitary equivalence.
Apply Corollary 1.6.15 to G1 ⊂ B1 (in place of F ⊂ A) and ε > 0, to obtain
G ⊂ B1 and δ1 (in place of the set G and the number δ, respectively in Corollary
1.6.15). Similarly, apply Corollary 1.6.15 to F1 ⊂ A (in place of F ⊂ A) and
ε > 0 to obtain F ⊂ A and δ2 (in place of the set G and the number δ in
Corollary 1.6.15).
Let δ = min(δ1 , δ2 ).
Suppose that the diagram
A
x
α1
B1
φ
−→
β
ց
ψ
−→
′
A
x
α2
B2
is the one constructed in Lemma 1.6.24 with homotopy path Ψ ∈
Map(B1 , B2 [0, 1]) between β ◦ α1 and ψ, and homotopy path Φ ∈
Map(A, A′ [0, 1]) between α2 ◦ β and φ, corresponding to the sets G ⊂ B1 ,
F ⊂ A and the number δ > 0.
Regarding the homotopy path Ψ as the homotopy path φt in Corollary 1.6.15,
we can obtain η1 , L1 as the numbers η and L in Corollary 1.6,15. Similarly,
replacing the above Ψ by Φ, we obtain η2 , L2 .
Documenta Mathematica 7 (2002) 255–461
316
Guihua Gong
Notice that the injectivity of α1 implies that, for each block B1j of B1 , SP(α1j ) =
Yj (= SP(B1j )). Therefore there is an η2 -dense subset {x1 , x2 , · · · xm } of X such
Sm
that i=1 SPα1j |xi is η1 -dense in Yj for each j ∈ {1, 2, · · · , s}.
Define λ1 : A(= MN (C(X))) → MmN (B2 ) by
λ1 (f ) = diag(1B2 ⊗ f (x1 ), 1B2 ⊗ f (x2 ), · · · , 1B2 ⊗ f (xm )).
Then λ1 ◦ α1 : B1 → MmN (B2 ) isSa homomorphism
defined by the point
Sm
s
evaluations on the η1 -dense subset j=1 i=1 SPα1j |xi ⊂ SPB1 . Also (α2 ⊗
idmN ) ◦ λ1 : A → MmN (A′ ) is defined by point evaluations on the η2 -dense
subset {xj }m
j=1 ⊂ X.
Let L = max(L1 , L2 ) and n = mN L. Define λ : A → Mn (B2 ) =
ML (MmN (B2 )) by λ = diag(λ1 , λ1 , · · · , λ1 ).
{z
}
|
L
Then, obviously, λ◦α1 : B1 → Mn (B2 ) satisfies the condition for λ in Corollary
1.6.15, for the homotopy Ψ, positive integer L1 , and η1 > 0. And so does
(α2 ⊗ idn ) ◦ λ : A → Mn (A′ ) for Φ, L2 and η2 .
Therefore, there are unitaries u1 ∈ Mn+1 (B2 ) and u2 ∈ Mn+1 (A′ ) such that
k((β ⊕ λ) ◦ α1 )(f ) − u1 (ψ ⊕ (λ ◦ α1 ))(f )u∗1 k < ε, ∀f ∈ G1 ,
k(φ ⊕ ((α2 ⊗ idn ) ◦ λ))(f ) − u2 ((α2 ⊗ idn+1 ) ◦ (β ⊕ λ))(f )u∗2 k < ε, ∀f ∈ F1 .
In the diagram in Lemma 1.6.24, if we replace B2 by Mn+1 (B2 ), A′ by
Mn+1 (A′ ), ψ by Adu1 ◦ (ψ ⊕ (λ ◦ α1 )), β by β ⊕ λ, α2 by Adu2 ◦ (α2 ⊗ idn+1 ),
and finally φ by φ ⊕ ((α2 ⊗ idn ) ◦ λ), then we have the diagram
A
x
α1
B1
φ
−→
β
ց
ψ
−→
′
A
x
α2
B2
for which, the lower left triangle is approximately commutative on G1 to within
ε and the upper right triangle is approximately commutative on F1 to within
ε. Since G1 and F1 are arbitrary finite subsets, we proved the following main
factorization result.
Ls
Theorem 1.6.26. Let B1 = j=1 Mk(j) (C(Yj )), where Yj are spaces: {pt},
∞
2
[0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S . Let X be a connected finite simplicial complex and let A = MN (C(X)).
Let α1 : B1 → A be an injective homomorphism. For any finite sets G ⊂ B1
and F ⊂ A, and for any numbers ε > 0 and δ > 0 there exists a diagram
A
x
α1
B1
φ
−→
β
ց
ψ
−→
′
A
x
α2
B2 ,
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
317
where
A′ = ML (A), and B2 is a direct sum of matrix algebras over the spaces: {pt},
∞
2
[0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S ;
ψ is an injective homomorphism, α2 is a unital injective homomorphism, and
φ is a unital simple embedding (see 1.6.17).
β ∈ Map(A, B2 ) is F -δ multiplicative.
Moreover
kψ(f ) − (β ◦ α1 )(f )k < ε, ∀f ∈ G;
kφ(f ) − (α2 ◦ β)(f )k < ε, ∀f ∈ F.
Corollary 1.6.27. Theorem 1.6.26 still holds if one replaces the injectivity
condition of α1 by the following condition:
For each block B1j of B1 , either α1j is injective or α1j (B1j ) is a finite dimensional
subalgebra of A.
(Of course, one still needs to assume that α1 (1B j ) 6= 0 for each block B1j of
1
B1 and that at least one block of B2 has spectrum different from the space of
single point (equivalently, X 6= {pt}) , if he wants the homomorphism ψ to be
injective.
If one does not assume the above condition, he could still get the following
dichotomy condition for ψ: For each block B1j of B1 and B2k of B2 , either ψ j,k
is injective or ψ j,k has a finite dimensional image.)
Proof: Write B1 = B ′ ⊕ B ′′ such that α1 is injective on B ′ and α1 (B ′′ ) ⊂ A is
of finite dimension.
Consider the finite dimensional algebra
D :=
M
B1i ⊂B ′
| )
(α1 (1B1i ) · C
M
α1 (B ′′ ) ⊂ A.
By Lemma 1.6.8, if β : A → B2 is sufficiently multiplicative, then β is close
to such a map β ′ that the restriction β ′ |D is a homomorphism. β ′ can be
connected to the original β by a linear path. If the original map β is sufficiently
multiplicative, then the connecting path, regarded as a map from A to B2 [0, 1],
is F -δ multiplicative for any pre-given finite set F ⊂ A and number δ > 0.
Therefore, with out loss of generality, we assume that β|D is a homomorphism
for the original map β in 1.6.24.
By Lemma 1.6.8 again, if Ψ : B1 → B2 [0, 1] is sufficiently multiplicative,
then Ψ is close to a map Ψ′ such that Ψ′ |r(B1 ) is a homomorphism, where
r(B1 ) is defined in 1.1.7(h). Note that Ψ|1 = ψ is a homomorphism and
(Ψ|0 )|r(B1 ) = β|D ◦ (α1 |r(B1 ) ) is also a homomorphism. From the proof of
Lemma 1.6.8, we can see that the above Ψ′ can be chosen such that Ψ′ |1 = Ψ|1
and Ψ′ |0 = Ψ|0 . Therefore, without loss of generality, we can assume that the
homotopy path Ψ in Lemma 1.6.24 satisfies that Ψ|r(B1 ) is a homomorphism.
Documenta Mathematica 7 (2002) 255–461
318
Guihua Gong
Up to a unitary equivalence, we can further assume that Ψt (1B1i ) = Ψt′ (1B1i )
for any t, t′ ∈ [0, 1] and any block B1i of B1 .
One can repeat the procedure in 1.6.25 to construct the homomorphism
λ : A → Mn (A), defined by point evaluations on an η2 -dense subset
{x1 , x2 , · · · , xm } ⊂ X, to satisfy the condition that λ ◦ α1j is defined by point
Sm
evaluations on an η1 -dense subset i=1 SPα1j |xi ⊂ SPB1j of sufficiently large
size, for each block B1j of the part B ′ . As in 1.6.25, we can define newβ to
be β ⊕ λ. At the same time, φ and α2 can also be defined as in 1.6.25. To
define ψ, we need to consider two cases. For the blocks B1j in B ′ , ψ can be
defined as in 1.6.25, since λ ◦ α1j is defined by point evaluations on an η1 -dense
subset (of sufficiently large size). For the blocks B1j ⊂ B ′′ , we define ψ to be
(β ⊕ λ) ◦ α1j = (newβ) ◦ α1j . (Note that β|α1 (B ′′ ) is a homomorphism.)
⊔
⊓
Remark 1.6.28. Once the diagram in Theorem 1.6.26 (or Corollary 1.6.27)
exists for A′ = ML (A), then for any L′ > L, one can construct a diagram with
the same property as in the theorem or the corollary with A′ = ML′ (A). This
is easily seen from the following.
| ) and r : A → r(A) be as in 1.1.7(h). Let newB2 =
Let r(A) = MN (C
oldB2 ⊕ r(A), newβ = oldβ ⊕ r, newφ = diag(oldφ, i ◦ r, · · · , i ◦ r ), newψ =
{z
}
|
L′ −L
oldψ ⊕ (r ◦ α1 ), and newα2 = oldα2 ⊕ diag(i, · · · , i), where i : r(A) → A ⊂
| {z }
L′ −L
M
(A) ⊂ M (A) is the inclusion (note that r(A) is a subalgebra of A as in
1.1.7(h)), and oldB2 , oldβ, oldφ and oldα2 are B2 , β, φ and α2 , respectively,
from Lemma 1.6.26 or Corollary 1.6.27.
L′ −L
L′
Ls
Corollary 1.6.29. Let B1 = j=1 Mk(j) (C(Yj )), where Yj are spaces: {pt},
Lt
∞
2
[0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S . Let A =
j=1 Ml(j) (C(Xj )),
where Xj are connected finite simplicial complex.
Let α1 : B1 → A be a homomorphism satisfying the following condition:
For each pair of blocks B1i of B1 and Aj of A, either the partial map α1i,j is
injective or α1i,j (B1i ) is a finite dimensional subalgebra of Aj .
For any finite sets G ⊂ B1 and F ⊂ A, and for any numbers ε > 0 and δ > 0,
there exists a diagram
A
x
α1
B1
φ
−→
β
ց
ψ
−→
′
A
x
α2
B2
where
A′ = ML (A), and B2 is a direct sum of matrix algebras over the spaces: {pt},
∞
2
[0, 1], S 1 , {TII,k }∞
k=2 , {TIII,k }k=2 , and S ;
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
319
ψ is a homomorphism, α2 is a unital injective homomorphism, and φ is a unital
simple embedding (see 1.6.17).
β ∈ Map(A, B2 ) is F -δ multiplicative.
Moreover,
kψ(f ) − (β ◦ α1 )(f )k < ε, ∀f ∈ G;
kφ(f ) − (α2 ◦ β)(f )k < ε, ∀f ∈ F.
If we further assume that α1 satisfies the condition that α1i,j (1B1i ) 6= 0 ∈ Aj for
any partial map α1i,j : B1i → Aj of α1 , then either the homomorphism ψ could
be chosen to be injective, or the spectra of all blocks of B2 could be chosen to
be the spaces of a single point.
Proof: We can construct the diagram for each block Aj of A, then put them
together in the obvious way. Using Remark 1.6.28, we can assume for each
block Aj , A′j = ML (Aj ) for the same L.
⊔
⊓
The following is Lemma 4.6 of [G4] (see Lemma 1.2 of [D2]).
Lt
Lemma 1.6.30. Let A =
i=1 Ml(i) (C(Xi )), where Xi are connected finite
simplicial complexes. Let A′ = ML (A). Let the algebra r(A) and the homomorphism r : A → r(A) be as in 1.1.7(h). Let B be a direct sum of matrix algebras over finite simplicial complexes of dimension at most m. Let φ : A → A ′
be a unital simple embedding (see Definition 1.6.17). For any (not necessarily
unital) (m · L)-large homomorphism φ′ : A → B, there is a homomorphism
λ : A′ ⊕ r(A) → B such that φ′ is homotopic to λ ◦ (φ ⊕ r).
Furthermore, λ could be chosen to satisfy the condition that for any block B j
with SP(B j ) 6= {pt}, the partial map λ·,j : A′ ⊕ r(A) → B j of λ is injective as
remarked in 1.6.21.
(This lemma will be applied in conjunction with Lemma 1.6.26 or Corollary
1.6.29. From here, one can see the importance of the requirement that φ is a
unital simple embedding.)
Remark 1.6.31. In order to apply Lemma 1.6.30 later, we would like to do
one more modification for Corollary 1.6.29. Let r : A → r(A) be as in 1.1.7(h).
Then the diagram in Corollary 1.6.29 could be modified to the following diagram
φ⊕r
A
−→
A′ x
⊕ r(A)
x
β⊕r
ց
α1
α2 ⊕id
B1
ψ⊕(r◦α1 )
−→
B2 ⊕ r(A)
which satisfies that β ⊕ r is F -δ multiplicative and
k(ψ ⊕ (r ◦ α1 ))(f ) − ((β ⊕ r) ◦ α1 )(f )k < ε, ∀f ∈ G;
k(φ ⊕ r)(f ) − ((α2 ⊕ id) ◦ (β ⊕ r))(f )k < ε, ∀f ∈ F.
Documenta Mathematica 7 (2002) 255–461
320
Guihua Gong
In the application of 1.6.29 and 1.6.30 in the proof of our main reduction
theorem, we will still denote B2 ⊕ r(A) by B2 , β ⊕ r by β, ψ ⊕ (r ◦ α1 ) by ψ,
and α2 ⊕ id by α2 . So the diagram is
A
x
α1
B1
2
φ⊕r
−→ A′ ⊕xr(A)
β
ց
α2
ψ
−→
B2 .
Spectral Multiplicity
In this section, we will show how to perturb a homomorphism φ : C(X) →
P Mk (C(Y ))P in such a way that the resulting homomorphism does not have
large spectral multiplicities (see 1.2.4). Namely, the following result will be
proved.
Theorem 2.1. Let X and Y be connected simplicial complexes with X 6= {pt}.
Set dim(X) + dim(Y ) = l. For any given finite set G ⊂ C(X), any ε > 0, and
any unital homomorphism φ : C(X) → P M• (C(Y ))P , where P ∈ M• (C(Y ))
is a projection, there is a unital homomorphism φ′ : C(X) → P M• (C(Y ))P
such that
(1) kφ(g) − φ′ (g)k < ε for all g ∈ G, and
(2) φ′ has maximum spectral multiplicity at most l.
| ))1 = F k X. The space
2.2. Let k be a positive integer. Let Hom(C(X), Mk (C
k
F X is compact and metrizable. We can endow the space F k X with a fixed
metric d as below.
Choose a finite set {fi }ni=1 ⊂ C(X) which generates C(X) as a C ∗ -algebra (e.g.
one can embed X into IRn , then choose {fi } to be the coordinate functions).
For any φ, ψ ∈ F k X which, by definition, are unital homomorphisms from
| ), define
C(X) to Mk (C
d(φ, ψ) =
n
X
i=1
kφ(fi ) − ψ(fi )k.
Without loss of generality, we can assume that the above finite set {fi }ni=1 ⊂ G.
On the other hand, F k X is a finite simplicial complex (see [DN], [Se] and [Bl1]).
2.3. Let k = rank(P ), where P is the projection in Theorem 2.1.
| ) such that P (y) = uy diag(1k , 0)u∗
For any fixed y, there is a unitary uy ∈ M• (C
y
| )P (y) with Mk (C
| )
(as in 1.2.1). Using this unitary, one can identify P (y)M• (C
| )P (y) to the element in Mk (C
| ) corresponding to
by sending a ∈ P (y)M• (C
| )P (y), the
the upper left corner of u∗y auy . (Notice that for any a ∈ P (y)M• (C
matrix
u∗y auy = u∗y P (y)aP (y)uy = u∗y P (y)uy u∗y auy u∗y P (y)uy
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
321
= diag(1k , 0)u∗y auy diag(1k , 0)
has the form
µ
(∗)k×k
0
0
0
¶
.)
| )P (y))1 can be
In this way, for any y ∈ Y , the space Hom(C(X), P (y)M• (C
identified with F k X.
Consider the disjoint union
[
| )P (y))1
Hom(C(X), P (y)M• (C
y∈Y
| )) × Y with the induced topology. Using the
as a subspace of Hom(C(X), M• (C
above identification we can define a locally trivial fibre bundle
[
| )P (y))1
Hom(C(X), P (y)M• (C
y∈Y
↓π
Y
with fibre F k X, as shown below, where π is the natural map sending any
|
element in the set Hom(C(X),
[ P (y)M• (C)P (y))1 to the point y.
| )P (y))1 .
For simplicity, write EP :=
Hom(C(X), P (y)M• (C
y∈Y
For any point y0 ∈ Y , there are an open set U ∋ y, and a continuous unitary
| ) such that P (y) = u(y)diag(1k , 0)u∗ (y). (See
valued function u : U → M• (C
| ) → Mk (C
| ) be the map taking any element in M• (C
| ) to
1.2.1.) Let R : M• (C
the k × k upper left corner of the element. Let the trivialization tU : π −1 (U ) →
| )P (y))1 ⊂
U × F k X be defined as follows. For any φ ∈ Hom(C(X), P (y)M• (C
π −1 (U ), where y ∈ U , define tu (φ) = (y, ψ), where ψ ∈ F k X is defined by
ψ(f ) = R(u∗ (y)φ(f )u(y)) for any f ∈ C(X).
µ
¶
(∗)k×k 0
∗
(Again, u (y)φ(f )u(y) is of the form
.)
0
0
Since the set Y is compact, there is a finite cover U = {U } of Y with the above
trivialization for each U . This defines a fibre bundle π : EP → Y .
(See §1.4 for the definition and other materials of fibre bundle.)
2.4. In the above fibre bundle, the structure group Γ ⊂ Homeo(F k X) could
be chosen to be the collection of all γ ∈ Homeo(F k X) of the form: there is a
| ) such that
unitary u ∈ Mk (C
γ(φ)(f ) = u∗ φ(f )u
for any φ ∈ F k X and f ∈ C(X).
One can see this as follows.
Documenta Mathematica 7 (2002) 255–461
322
Guihua Gong
Suppose that U and V are two open sets in U, and tU and tV are trivializations,
| ) and v : V →
as in 2.3, defined by unitary valued functions u : U → M• (C
| ), respectively.
M• (C
k
k
For any point y ∈ U ∩ V , the map tU ◦ t−1
V : F X → F X, can be computed
as below.
| ) by
For any φ ∈ F k X, define φ̃ : C(X) → M• (C
µ
¶
φ(f )k×k 0
φ̃(f ) =
, ∀f ∈ C(X).
0
0
Then
and
∗
|
t−1
V (φ)(f ) = v(y)φ̃(f )v (y) ∈ P (y)M• (C)P (y),
∗
∗
tU ◦ t−1
V (φ)(f ) = R(u (y)v(y)φ̃(f )v (y)u(y)).
Notice that
u(y)diag(1k , 0)u∗ (y) = P (y) = v(y)diag(1k , 0)v ∗ (y).
It follows that v ∗ (y)u(y) commutes with diag(1k , 0). This implies that this
matrix has the form
¶
µ
(w1 )k×k 0
,
0
w2
where both w1 and w2 are unitaries. This shows that
∗
k
tU ◦ t−1
V (φ)(f ) = w1 φ(f )w1 , ∀φ ∈ F X, f ∈ C(X).
In other words, tU ◦ t−1
V ∈ Γ.
Obviously, Hom(C(X), P M• (C(Y ))P )1 can be regarded as a collection of continuous cross sections of the bundle π : EP → Y .
| ) and unitary u ∈ Mk (C
| ),
Since for any elements a, b ∈ Mk (C
kuau∗ − ubu∗ k = ka − bk,
it is easy to see that the metric d on F k X defined in 2.2 is invariant under
the action of any element in Γ in the sense of 1.4.6.
2.5. There is a natural map
θ : F k X −→ P k X
| ), define
defined as follows. For any φ ∈ F k X given by φ : C(X) → Mk (C
θ(φ) = SP(φ) ∈ P k X,
counting multiplicities. (See 1.2.5 and 1.2.7.)
For each point x = [x1 , x2 , · · · , xk ] ∈ P k X, if the element xi appears µi times
in x for i = 1, 2, · · · , k, then the maximum multiplicity of x is defined to be
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
323
the maximum of µ1 , µ2 , · · · , µk . The maximum multiplicity of a point φ ∈ F k X
is defined to be the maximum multiplicity of θ(φ) ∈ P k X, which agrees with
| ) defined in
the maximum multiplicity of homomorphism φ : C(X) → Mk (C
1.2.4.
The homomorphism φ ∈ Hom(C(X), P M• (C(Y ))P )1 corresponds to a continuous cross section f : Y → EP . This correspondence is one to one. For any
cross section f : Y → EP , any point y ∈ Y , the maximum multiplicity of f (y)
is understood to be that obtained by regarding f (y) as an element in F k X by
| )P (y))1 with F k X. Note that the
an identification of Hom(C(X), P (y)M• (C
maximum multiplicity of an element φ ∈ F k X is invariant under the action of
any element of Γ.
2.6. It is easy to see that for any finite set G ⊂ C(X) and ε > 0, there is
an ε′ > 0 such that if d(φy , φ′y ) < ε′ for any y ∈ Y , then kφ(g) − φ′ (g)k < ε
for any g ∈ G, where φy , φ′y ∈ F k X are determined by an identification of
| )P (y))1 with F k X, as above. (Again the choice of the
Hom(C(X), P (y)M• (C
identification is not important, because the metric d is invariant under the
action of any element in Γ.)
Before proving Theorem 2.1, we prove the following weak version of Theorem
2.1.
Lemma 2.7. Let X and Y be as in Theorem 2.1. Let k > l = dim(X)+dim(Y ).
For any given finite set G ⊂ C(X), any ε > 0, and any unital homomorphism φ : C(X) → P M• (C(Y ))P , where P ∈ M• (C(Y )) is a projection with
rank(P ) = k, there is a unital homomorphism φ′ : C(X) → P M• (C(Y ))P
such that
(1) kφ(g) − φ′ (g)k < ε for all g ∈ G, and
(2) φ′ has maximum spectral multiplicity at most k − 1.
Comparing with Theorem 2.1, in the above result, we allow the maximum
spectral multiplicity of the resulting homomorphism to be larger than l— only
require it to be smaller than k = rank(P )—the maximum possible multiplicity.
Since we assume that all the generators fi of C(X) are inside the set G, Lemma
2.7 is equivalent to the following theorem.
Lemma 2.8. Suppose that X, Y , and P are as in Theorem 2.1 and that
rank(P ) = k > dim(X) + dim(Y ). For any ε > 0 and any cross section
f : Y → EP , there is a cross section f ′ : Y → EP such that
(1) d(f (y), f ′ (y)) < ε for all y ∈ Y , and
(2) f ′ (y) has multiplicity at most k − 1 for all y ∈ Y .
To prove our main theorem of this section—Theorem 2.1, we need the following
result. The proof of this result will be given after the proof of Theorem 2.1.
Theorem 2.9. Suppose that X is a connected simplicial complex and X 6=
{pt}. For any ε > 0 and any x ∈ F m X, there is a contractible open neighborDocumenta Mathematica 7 (2002) 255–461
324
Guihua Gong
hood Ux ∋ x with Ux ⊂ Bε (x) ⊂ F m X such that
πi (Ux \{x}) = 0
for any 0 ≤ i ≤ m − 2. In other words, F m X has property D(m) as in 1.4.3.
We will use Theorem 2.9 and Corollary 1.4.10 (see also Theorem 1.4.9) to prove
the following relative version of Lemma 2.8 (which gives rise to Lemma 2.8, by
taking Y1 = ∅ ).
Lemma 2.10. Let X, Y , and P be as in Theorem 2.1, and Y1 ⊂ Y be a sub
simplicial complex. Suppose that rank(P ) = k > dim(X) + dim(Y ). Suppose
that a cross section f : Y → EP satisfies the condition that f (y) has multiplicity
at most k − 1, for any y ∈ Y1 . It follows that for any ε > 0, there is a cross
section f ′ : Y → EP such that
(1) d(f (y), f ′ (y)) < ε for all y ∈ Y , and
(2) f ′ (y) has multiplicity at most k − 1 for all y ∈ Y .
(3) f ′ (y) = f (y), for any y ∈ Y1 .
Proof: Let F1 ⊂ F k X denote the subset of all elements of maximum multiplicity equal to k. In other words, a homomorphism in F1 has one dimensional
| ))1
range. Obviously, F1 is the set of all homomorphisms φ ∈ Hom(C(X), Mk (C
which are of the form
f (x)
f (x)
φ(f ) =
..
.
f (x)
for a certain point x ∈ X. Hence F1 is homeomorphic to X, and dim(F1 ) =
dim(X).
As mentioned in 2.5, the maximum multiplicity of an element of F k X is invariant under the action of Γ. So F1 is an invariant subset under the action of
Γ.
The conclusion of the Lemma 2.10 follows from Corollary 1.4.10 with EP → Y ,
F k X, F1 , Y1 and k in place of M → X, F , F1 , X1 and m in Corollary 1.4.10,
respectively. (Note that, from Theorem 2.9, F k X has property D(k).)
⊔
⊓
The above lemma is equivalent to the following lemma (we stated it with projection Q instead of P to emphasis that we may use projections other than
P —we will use subprojections of P ).
Lemma 2.11. Let Y2 ⊂ Y1 ⊂ Y be sub-simplicial complexes of Y . Let Q ∈
M• (C(Y1 )) be a projection with rank m > l = dim(X) + dim(Y ). For any given
finite set G ⊂ C(X), any ε > 0, and any unital homomorphism ψ : C(X) →
QM• (C(Y1 ))Q with the property that for any y ∈ Y2 , the multiplicity of ψ at y
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
325
is at most m − 1, there is a unital homomorphism ψ ′ : C(X) → QM• (C(Y1 ))Q
such that
(1) kψ(g)(y) − ψ ′ (g)(y)k < ε for all g ∈ G and y ∈ Y1 ,
(2) ψ ′ has spectral multiplicity at most m − 1,
(3) ψ ′ |Y2 = ψ|Y2 .
In the proof of Theorem 2.1, we will not use Theorem 2.9 or Lemma 2.10
directly. We will use Lemma 2.11 instead. (So we do not need anything from
fibre bundles in the rest of the proof of Theorem 2.1.)
2.12 Sketch of the idea of the proof of Theorem 2.1. Note that
the proof of Lemma 2.10 can not be used to prove Theorem 2.1 (or the fibre
bundle version of Theorem 2.1) in a straightforward way. For example, if we
let F1 ⊂ F k X be the subset of all elements with maximum multiplicity at
least l + 1 (instead of k), then dim(F1 ) may be very large— much larger than
dim(X). In fact, dim(F1 ) also depends on k and l.
In Lemma 2.11 (or Lemma 2.10), we have already perturbed the homomorphism
to avoid the largest possibility of maximum multiplicity—k. Next, we will
perturb it again to avoid the next largest possibility of maximum multiplicity—
k − 1. We will continue the procedure in this way.
In general, suppose that the homomorphism φ : C(X) → P M• (C(Y ))P has
maximum multiplicity m, with l < m ≤ k, we will prove that φ can be approximated arbitrarily well by another homomorphism φ′ with maximum multiplicity at most m − 1. Once this is done, Theorem 2.1 follows from a reverse
induction argument beginning with m = k > l. (Note that for the case k ≤ l,
we have nothing to prove.)
To do the above, we need to work simplex by simplex. In fact, on each small
simplex, the homomorphism
φ can be decomposed into a direct sum of several
L
homomorphisms
i φi , such that the projections φi (1) has rank at most m.
Then we can apply Lemma 2.11 to each φi to avoid maximum multiplicity m.
With these ideas in mind, it will not be difficult for the reader to construct the
proof of Theorem 2.1. The complete detail will be contained in the next few
lemmas, in particular, see the proof of Lemma 2.16.
Lemma 2.13. Suppose that P, X and Y are as in Theorem 2.1. For any
ε > 0 and any positive integer d, there is a δ > 0 such that if ∆ ⊂ Y is a
simplex of dimension d, if P is regarded as a projection in M• (C(∆)), and if
ψ : C(X) → P M• (C(∂∆))P is a homomorphism such that
kψ(g)(y) − ψ(g)(y ′ )k < δ,
∀g ∈ G, y, y ′ ∈ ∂∆,
then there is a homomorphism ψ ′ : C(X) → P M• (C(∆))P such that
(1) kψ ′ (g)(y) − ψ ′ (g)(y ′ )k < ε, ∀g ∈ G, y, y ′ ∈ ∆, and
(2) ψ ′ |∂∆ = ψ.
Proof: Note that P |∆ is a trivial projection, since a simplex ∆ is contractible.
Documenta Mathematica 7 (2002) 255–461
326
Guihua Gong
So P M• (C(∆))P ∼
= Mk (C(∆)), where k = rank(P ). The lemma follows from
the fact that
| ))1
F k X = Hom(C(X), Mk (C
is a simplicial complex (see [DN] and [Bl]), which is locally contractible (see
1.4.2 and 1.4.3).
⊔
⊓
We need the following lemma, which is obviously true.
Lemma 2.14. Suppose that φ : C(X) → P M• (C(Y ))P has maximum spectral
multiplicity at most m. Then there exist η > 0 and δ > 0 such that the following
statement holds.
For any subset Z ⊂ Y with diameter(Z) < η, and homomorphism ψ : C(X) →
P M• (C(Z))P with the property that
kψ(g)(z) − φ(g)(z)k < δ, ∀z ∈ Z, g ∈ G,
there is a decomposition of ψ such as described below.
There are open sets O1 , O2 , · · · , Ot ⊂ X, with mutually disjoint closures
(i.e., Ōi ∩ Ōj = ∅, ∀i 6= j), and there are mutually orthogonal projections
Q1 , Q2 , · · · , Qt ∈ M• (C(Z)) and homomorphisms ψi : C(X) → Qi M• (C(Z))Qi
such that
Pt
1. ψ = i=1 ψi
Pt
2. P (z) = i=1 Qi (z), ∀z ∈ Z,
3. rank(Qi ) ≤ m, and
4. SPψi ⊂ Oi for all i.
Proof: One can prove it using the following fact. Suppose that SPφ ⊂ ∪Oi . If
ψ is close enough to φ, then SPψ ⊂ ∪Oi and #(SPφ ∩ Oi ) = #(SPψ ∩ Oi ) (see
1.2.12).
Qi in the lemma should be chosen to be the spectral projections of ψ corresponding to the open sets Oi (see 1.2.9).
(Notice that Y is compact and that G ⊂ C(X) contains a set of the generators.)
⊔
⊓
Lemma 2.15. Suppose that φ : C(X) → P M• (C(Y ))P has maximum spectral
multiplicity at most m. Then there exist η > 0 and δ > 0 such that the following
statement holds.
For any ε > 0, any simplex ∆ ⊂ Y (of any simplicial decomposition of Y ) with
diameter(∆) < η, and any homomorphism ψ : C(X) → P M• (C(∆))P with the
following properties:
(i) kψ(g)(z) − φ(g)(z)k < δ, ∀z ∈ ∆, g ∈ G, and
(ii) ψ|∂∆ has maximum multiplicity at most m − 1,
there exists a homomorphism ψ ′ : C(X) → P M• (C(∆))P such that
(1) kψ(g)(y) − ψ ′ (g)(y)k < ε for all g ∈ G and y ∈ ∆;
(2) ψ ′ has spectral multiplicity at most m − 1;
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
327
(3) ψ ′ |∂∆ = ψ|∂∆ .
Proof: . Suppose that η and δ are as in Lemma 2.14. If ψPis as described in
t
this lemma, then one can obtain the decomposition ψ = i=1 ψi of ψ as in
Lemma 2.14.
Then we only need to apply Lemma 2.11 to each map ψi to obtain ψi′ : C(X) →
Qi M• (C(Z))Qi to satisfy the conclusion of Lemma 2.11 with ψi , ∆, ∂∆, and
Qi in place of ψ, Y1 , Y2 , and Q, respectively.
′
If ε is small enough, then SPψ
Pt i ⊂ ′Oi , where the open sets Oi are from Lemma
′
2.14. Hence the sum ψ = i=1 ψi is as desired.
⊔
⊓
Lemma 2.16. Suppose that φ : C(X) → P M• (C(Y ))P has maximum spectral
multiplicity at most m > l = dim(X) + dim(Y ). For any simplicial subcomplex
Y1 ⊂ Y , with respect to any simplicial decomposition of Y , and any ε > 0, there
is a homomorphism φ′ : C(X) → P M• (C(Y1 ))P with multiplicity at most m−1
such that
kφ′ (g)(y) − φ(g)(y)k < ε, ∀g ∈ G, y ∈ Y1 .
(In particular, the above is true for Y1 = Y .)
Proof: We will prove the lemma by induction on dim(Y1 ).
If dim(Y1 ) = 0, the lemma follows from the fact that, for a connected simplicial
complex X with X 6= {pt}, the subset of homomorphisms with distinct spectrum (maximum spectral multiplicity one) is dense in
| ))1 .
Hom(C(X), Mk (C
Suppose that the lemma is true for any simplicial subcomplex of dimension n,
with respect to any simplicial decomposition.
Let Y1 ⊂ Y be a simplicial complex of dimension n + 1 ≤ dim Y , with respect
to some simplicial decomposition of Y .
Let ε > 0.
Let δ1 , η1 be the δ and η of Lemma 2.15.
Apply Lemma 2.13 with n + 1 in place of d, and 14 min(ε, δ1 ) in place of ε, to
find δ2 as δ in the lemma.
Choose η2 > 0 such that if dist(y, y ′ ) < η2 , then
(∗)
kφ(g)(y) − φ(g)(y ′ )k <
1
min(ε, δ1 , δ2 ), ∀g ∈ G.
4
Endow Y1 with a simplicial complex structure such that diameter(∆) <
min(η1 , η2 ) for any simplex ∆ of Y1 . Let Y ′ ⊂ Y1 be the n-skeleton of Y1
with respect to the simplicial structure.
From the inductive assumption, there is a homomorphism φ1 : C(X) →
P M• (C(Y ′ ))P , with multiplicity at most m − 1, such that
(∗∗)
kφ1 (g)(y) − φ(g)(y)k <
1
min(ε, δ1 , δ2 ), ∀g ∈ G and y ∈ Y ′ .(∗∗)
4
Documenta Mathematica 7 (2002) 255–461
328
Guihua Gong
Consider a fixed simplex ∆ ⊂ Y1 of top dimension (i.e., dim(∆) = n + 1). Let
us extend φ1 |∂∆ to ∆ (notice that ∂∆ ⊂ Y ′ ).
For any two points y, y ′ ∈ ∂∆, applying (*) to the pair of points y and y ′ ,
applying (**) to the points y and y ′ separately, and combining all these three
inequalities together, we get
kφ1 (g)(y) − φ1 (g)(y ′ )k <
3
min(ε, δ1 , δ2 ) < δ2 , ∀g ∈ G.
4
By Lemma 2.13 and the way δ2 is chosen, there is a homomorphism (let us
still denote it by φ1 ) φ1 : C(X) → P M• (C(∆))P , which extends the original
φ1 |∂∆ , such that
(∗∗∗)
kφ1 (g)(y) − φ1 (g)(y ′ )k <
1
min(ε, δ1 ), ∀g ∈ G and y, y ′ ∈ ∆.
4
For any point y ∈ ∆, choose a point y ′ ∈ ∂∆. Applying both (*) and (***)
to the pair (y, y ′ ), applying (**) to the point y ′ ∈ ∂∆ ⊂ Y ′ and combining all
these three inequalities together, we get
(∗∗∗∗)
kφ1 (g)(y) − φ(g)(y)k <
3
min(ε, δ1 ), ∀g ∈ G and y ∈ ∆.
4
Since δ1 and η1 are chosen as in Lemma 2.15, and diameter(∆) < η1 , it follows
from (∗ ∗ ∗∗) and Lemma 2.15 that there is a homomorphism φ′ : C(X) →
P M• (C(∆))P such that
(1) kφ′ (g)(y) − φ1 (g)(y)k < 41 ε, ∀g ∈ G and y ∈ ∆.
(2) φ′ has spectral multiplicity at most m − 1.
(3) φ′ |∂∆ = φ1 |∂∆ .
Combining (1) above with (****), yields
kφ′ (g)(y) − φ(g)(y)k <
1
3
min(ε, δ1 ) + ε ≤ ε, ∀g ∈ G and y ∈ ∆.
4
4
Carry out the above construction independently for each simplex ∆. Since the
definition of φ′ on ∂∆ is as same as φ1 , the definitions of φ′ on different simplices
are agree on their intersection. By Lemma 1.2.14, this yields a homomorphism
over the whole set Y1 . The lemma follows.
⊔
⊓
Obviously, Theorem 2.1 follows from Lemma 2.16 by reverse induction argument beginning with m = k. (Note that we only need Lemma 2.16 for the case
Y1 = Y .)
Now we are going to prove Theorem 2.9, which is the only missing part in the
proof of Theorem 2.1. The proof is somewhat similar to the proof of Theorem
6.4.2 of [DN]. It will therefore be convenient to recall some of the terminology
and notation of [DN]. (It will be important to consider a certain method of
decomposing the space F k X.)
2.17.
Recall from 6.17 of [DN] (cf. 1.2.4 above) that there is a map λ :
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
329
X k × U (k) → F k X, defined as follows. If u ∈ U (k) and (x1 , x2 , · · · , xk ) ∈ X k ,
then
f (x1 )
f (x2 )
∗
(λ(x1 , x2 , · · · , xk , u))(f ) = u
u
.
..
f (xk )
for any f ∈ C(X). Since λ is surjective, F k X can be regarded as a quotient
space X k × U (k). Therefore, for convenience, a point in F k X will be written
as
[x1 , x2 , · · · , xk , u]
which means λ(x1 , x2 , · · · , xk , u).
With the above notation, it is easy to see that, if X is path connected and is
not a single point, then any element in F k X can be approximated arbitrarily
well by elements in F k X with distinct spectra.
2.18.
If X1 ⊂ X is a subset, then define F k X1 to be the subset of F k X
| )) with SP(φ) ⊂ X1
consisting of those homomorphisms φ ∈ Hom(C(X), Mk (C
as a set. Obviously, if X1 is open (closed resp.), then F k X1 is open (closed
resp.).
If X1 , X2 , · · · , Xi are disjoint subspaces of X, and k1 , k2 , · · · , ki are nonnegative
integers with
k1 + k2 + · · · + ki = k,
then define F (k1 ,k2 ,···,ki ) (X1 , X2 , · · · , Xi ) to be the subset of F k X consisting of
all φ with
#(SP(φ) ∩ Xi ) = ki
counting multiplicity.
Usually when we use the above notation, we suppose that X̄i1 ∩ X̄i2 = ∅ if
i1 6= i2 , where X̄i is the closure of Xi . In this case,
a
F k (X1 ∪ X2 ∪ · · · ∪ Xi ) =
F (k1 ,k2 ,···,ki ) (X1 , X2 , · · · , Xi )
k1 +k2 +···+ki =k
is a disjoint union of separate components.
2.19. For each i-tuple (k1 , k2 , · · · , ki ) with
k1 + k2 + · · · + ki = k,
| k ) to be the collection of i-tuples (p1 , p2 , · · · , pi )
one can define G(k1 ,k2 ,···,ki ) (C
| ) with rank(pj ) = kj (1 ≤ j ≤ i) and
of orthogonal projections pj ∈ Mk (C
Pi
|
| k
j=1 pj = 1 ∈ Mk (C ). Note that if i = 2, G(k1 ,k2 ) (C ) is the ordinary
| k ).
| k ) = Gk (C
complex Grassmannian manifold Gk1 (C
2
Documenta Mathematica 7 (2002) 255–461
330
Guihua Gong
For each fixed i-tuple (k1 , k2 , · · · , ki ), there is a locally trivial fibre bundle
F k1 (X1 ) × F k2 (X2 ) × · · · × F ki (Xi ) −→ F (k1 ,k2 ,···,ki ) (X1 , X2 , · · · , Xi )
↓
| k ).
G(k1 ,k2 ,···,ki ) (C
2.20.
For certain purposes, it is more convenient to use CW complexes
(instead of simplicial complexes).
For the terminology used below, see [Wh].
Suppose that (X, A) is a relative CW complex pair. If X is path connected,
then (X, A) is zero connected CW complex pair, no matter A is connected
or not. In particular, (X, A) is homotopy equivalent to (X1 , A), where X1 is
obtained from A by attaching finitely many cells of dimension ≥ 1 (see Theorem
2.6 of Chapter five of [Wh]). This can not be done if one only uses simplicial
complex pair. (Note, we always assume our CW complexes to be finite CW
complexes without saying so.)
For a relative CW complex pair (X, A), define FAm X ⊂ F m X to be the subspace consisting of those elements x ∈ F m X, with
SP(x) ∩ A 6= ∅.
(This is different from the set F m A (defined in 2.18) which consists of elements
x ∈ F m X such that SP(x) ⊂ A.)
Lemma 2.21. Suppose that (X, A) is a relative CW complex pair. Suppose
that X is obtained from A by attaching cells of dimension at least 1. It follows
that the inclusion
FAm X ֒→ F m X
is m − 1 equivalent, i.e., i∗ : πj (FAm X) → πj (F m X) is an isomorphism for
any 0 ≤ j ≤ m − 2 and a surjection for j = m − 1, where i∗ is induced by the
inclusion map.
The proof of this lemma is divided into two steps.
Lemma 2.22. Lemma 2.21 is true if X is obtained from A by attaching several
cells of dimension 1.
Proof: Let X = A∪e1 ∪e2 ∪· · ·∪et , where e1 , e2 , · · · , et are 1-cells with ∂ei ⊂ A.
Then F m X\FAm X consists of those points whose spectra are contained in
◦
◦
◦
e1 ∪ e2 ∪ · · · ∪ et ,
◦
where each ej = ej \∂ej is homeomorphic to (0, 1).
In other words,
a
◦
◦
◦
F m X\FAm X =
F (k1 ,k2 ,···,kt ) ( e1 , e2 , · · · , et ).
k1 +k2 +···+kt =m
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
331
P
For each fixed t-tuple (k1 , k2 , · · · , kt ) with
ki = m, the space
◦
◦
◦
(k1 ,k2 ,···,kt ) e1 e2
e
F
( , , · · · , t ) is a smooth manifold. To see this, we can
consider the fibre bundle
◦
◦
◦
◦
◦
◦
F k1 ( e1 ) × F k2 ( e2 ) × · · · × F kt ( et ) −→ F (k1 ,k2 ,···,kt ) ( e1 , e2 , · · · , et )
↓
| )
G(k1 ,k2 ,···,kt ) (C
introduced in 2.19. Evidently, the fibre of the bundle is
◦
◦
◦
2
2
2
F k1 ( e1 ) × F k2 ( e2 ) × · · · × F kt ( et ) ∼
= IRk1 × IRk2 × · · · × IRkt .
Note that the above fibre bundle has an obvious cross section (see [DN]). Therefore, the fibre bundle can be regarded as a smooth vector bundle with the vector
2
2
2
space IRk1 +k2 +···+kt as the fibre. The zero section of the bundle has codimension
k12 + k22 + · · · + kt2 ≥ k1 + k2 + · · · + kt = m.
By a standard argument from topology, using the transversality theorem, the
lemma can be proved. (See [DN, 6.3.4] for details.)
⊔
⊓
2.23. The next step is to prove Lemma 2.21 by induction, starting with 2.22.
Since the proof is a complete repetition of the proof of Theorem 6.4.2 of [DN],
we omit the detail—only point out how to define the collection W (n, r) in our
setting, and several small modifications.
Let W (1, 0) = {A}—the set with single element: the space A. For r >
0, W (n, r) is the class of all finite CW complexes, each of which is obtained by
attaching at most one cell of dimension n to a space in W (n, r − 1). Let
W (n + 1, 0) =
+∞
[
W (n, r).
r=0
Lemma 2.22 says that FAm X → F m X is m − 1 equivalent if X ∈ W (2, 0). In
applying the argument in [DN, 6.4.2], FAm X is in place of F k (X) and F m X is
in place of F k+1 (X). All the other parts of the proof follow from [DN]. The
only thing needs mentioning is that the inclusion
FAm (X\αI ) ֒→ FAm X
is 1-equivalent, where αI is a set of finitely many points inside one of the
n-cells of X, and n ≥ 2. To prove this statement, one needs to prove that
any continuous map from S 1 to FAm X can be perturbed to a map from S 1 to
FAm (X\αI ). To do this, he can first perturb a map to a piecewise linear map
for which the image will be one dimensional. And the resulting map can be
Documenta Mathematica 7 (2002) 255–461
332
Guihua Gong
easily perturbed again to a map whose spectrum avoids any given finite set of
points in any cell of dimension at least 2.
⊔
⊓
2.24. Let X be a simplicial complex, and x0 ∈ X be a vertex.
Define X ′ to be the sub-simplicial complex consisting of all the simplices ∆ (
and their faces) with ∆ ∋ x0 . Then X ′ is contractible.
We also use x0 to denote the point in F m X defined by
| )
φ(f ) = f (x0 ) · 1m ∈ Mm (C
for each f ∈ C(X).
We can easily prove the following claim: The map F m X ′ \{x0 } ֒→ F m X ′ is
(m − 1) equivalent.
To see this, let
A = ∪{∆ | ∆ is a simplex of X ′ and x0 ∈
/ ∆}.
Then A is a sub-simplicial complex of X ′ . A may not be connected, but (X ′ , A)
is 0-connected. In the notation of 1.4.2,
X ′ = Star({x0 })
and
A = Star({x0 })\Star({x0 }).
It is obvious that A ֒→ X ′ \{x0 } is a homotopy equivalence. Therefore,
FAm X ′ ֒→ (F m X ′ )\{x0 } is a homotopy equivalence. By Lemma 2.21, the claim
holds. In particular,
πi (F m X ′ \{x0 }) = 0
for any 0 ≤ i ≤ m − 2, since F m X ′ is contractible. Equivalently,
πi (F m (X ′ \A)\{x0 }) = 0
for any 0 ≤ i ≤ m − 2. Note that X ′ \A is an open neighborhood of x0 ∈ X,
which is the interior of X ′ .
2.25. Proof of Theorem 2.9. Suppose that
SP(x) = { λ1 , λ1 , · · · , λ1 , λ2 , λ2 , · · · , λ2 , · · · , λi , λi , · · · , λi },
|
{z
} |
{z
}
{z
}
|
k1
k2
ki
where λ1 , λ2 , · · · , λi ∈ X are distinct points and k1 + k2 + · · · + ki = m. Choose
mutually disjoint open sets U1 ∋ λ1 , U2 ∋ λ2 , · · · , Ui ∋ λi , in X. Then there is
a locally trivial fibre bundle
F k1 (U1 ) × F k2 (U2 ) × · · · × F ki (Ui ) −→ F (k1 ,k2 ,···,ki ) (U1 , U2 , · · · , Ui )
↓
| ).
G(k1 ,k2 ,···,ki ) (C
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
333
| ) = U (m)/(U (k1 ) × U (k2 ) × · · · × U (ki )) is a smooth
Note that G(k1 ,k2 ,···,ki ) (C
Pi
manifold of dimension t := m2 − j=1 kj2 . There is a small contractible neighborhood Ux ⊂ Bε (x) which is homeomorphic to the space
F k1 (X1 ) × F k2 (X2 ) × · · · F ki (Xi ) × IRt ,
where X1 , X2 , · · · , Xi are mutually disjoint open subsets of X. The space Xi
can be chosen so that X̄i is the simplicial complex X ′ as in 2.24 corresponding
to vertex λi , with respect to some simplicial decomposition of X.
The following fact is well known in topology. Suppose that X and Y are
connected CW complexes with base points x0 , y0 respectively. If X\{x0 } is
l1 -connected and X\{y0 } is l2 -connected, then (X × Y )\{x0 , y0 } is (l1 + l2 + 2)connected.
Combining this fact with 2.24, we conclude that Ux \{x} is
i
X
j=1
kj − 2 + t = m − 2 + t
connected. This ends the proof.
⊔
⊓
3
Combinatorial Results
In this section, we will prove certain results of a combinatorial nature, for the
preparation of the proof of the Decomposition Theorem—Theorem 4.1 of the
next section.
We will need the results in the case that certain multiplicities are general—not
just equal to one. For the sake of clarity, we will first state and prove the results
in the special case of multiplicity one. We will then consider the general case.
3.1. Suppose that X is a simplicial complex. Let σ denote the simplicial
complex structure of X—which tells what the simplices of X are, and what
the faces of each simplex of X are. In this section, we will use (X, σ) to denote
the simplicial complex X with simplicial structure σ, to emphasize that we
may endow the same space X with different simplicial complex structures.
In this section, we will reserve the notation, σ, τ, σ1 , τ1 , · · · , etc., for simplicial
complex structures.
Recall that, if ∆ is a simplex, its boundary is denoted by ∂∆. For example, if
dim(∆) = 0, i.e., ∆ = {pt}, the set consisting of a single point, then ∂∆ = ∅;
if dim(∆) = 1, i.e., ∆ is an interval, then ∂∆ is the set consisting of the two
extreme points of the interval. Let us also consider the set ∆\∂∆, and denote
it by interior(∆).
If (X, σ) is a simplicial complex, then for any point x ∈ X, there is a unique
simplex ∆ such that x ∈ interior(∆).
Documenta Mathematica 7 (2002) 255–461
334
Guihua Gong
As usual, if each simplex of (X, σ1 ) is a union of certain simplices of (X, σ2 ),
then we shall call σ2 a subdivision of σ1 . This is equivalent to the property
that any simplex of (X, σ2 ) is contained in a simplex of (X, σ1 ).
The notation Vertex(X, σ) (respectively, Vertex(∆) ) will be used to denote
the set of vertices of (X, σ) (or of the simplex ∆).
Definition 3.2. Let E = {1, 2, · · · , K} be an index set. (The index set E
can be any set with exactly K elements.) Let K1 , K2 , · · · , Km be non negative
integers with
K1 + K2 + · · · + Km = K.
A grouping of E of type (K1 , K2 , · · · , Km ) is a collection of m mutually
disjoint index sets E1 , E2 , · · · , Em with
E = E1 ∪ E2 ∪ · · · ∪ E m ,
and #(Ej ) = Kj for each 1 ≤ j ≤ m. (Cf. 1.5.18.)
Usually, we will keep the tuple (K1 , K2 , · · · , Km ) fixed and just call the collection E1 , E2 , · · · , Em a grouping of E (without mentioning the type).
(Most of the time, K1 , K2 , · · · Km will be positive integers, i.e., nonzero. But
for convenience, we allow some numbers Ki = 0, and then the corresponding
sets Ei should be the empty set.)
3.3.
Let (X, σ) be a simplicial complex. Suppose that, associated to
each vertex x ∈ X, there is a grouping E1 (x), E2 (x), · · · , Em (x) of E of type
(K1 , K2 , · · · , Km ). (In our application in the proof of Theorem 4.1, the index
set E will be the spectrum of a homomorphism at the given vertex, see 1.5.13,
1.5.17–1.5.22.)
Suppose that these groupings for all the vertices are chosen arbitrarily. Then,
in general, for a simplex ∆ with vertices x0 , x1 , · · · , xn , the intersections
\
Ej (x) = Ej (x0 ) ∩ Ej (x1 ) ∩ · · · ∩ Ej (xn ), j = 1, 2, · · · , m,
x∈Vertex(∆)
may have very few elements—the sets Ej (x0 ), Ej (x1 ), · · · , Ej (xn ) may be very
different.
The purpose of this section is to introduce a subdivision (X, τ ) of (X, σ), and
to associate to each new vertex of (X, τ ) a grouping to make the following true:
For any simplex ∆ of (X, τ ) (after the subdivision), for each j, the number of
elements in the intersection
\
Ej (x)
x∈Vertex(∆)
is not much less than the number of elements in each individual set Ej (x) (note
that #(Ej (x)) = Kj for each x); in other words, the groupings of adjacent
vertices (after subdivision) should be almost as the same as each other.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
335
First we will state the following lemma (the proof will be given in 3.15). Later
on, we will need a relative version of the lemma.
(See 1.5.17 to 1.5.23 for the explanations of the role of this lemma in §4. To
visualize the following lemma, see 1.5.21 for the explanation of the one dimensional case.)
Lemma 3.4. Let (X, σ) be a simplicial complex consisting of a single simplex
X and its faces. Suppose that associated to each x ∈ Vertex(X, σ), there is a
grouping E1 (x), E2 (x), · · · , Em (x) of E (of type (K1 , K2 , · · · , Km )).
It follows that there is a subdivision (X, τ ) of (X, σ), and associated to each
new vertex x ∈ Vertex(X, τ ), there is a grouping E1 (x), E2 (x), · · · , Em (x) of E
(of type (K1 , K2 , · · · , Km )), (for any old vertex of (X, σ), the grouping should
not be changed), such that the following hold.
For each newly introduced vertex x ∈ Vertex(X, τ ),
\
(1)
Ej (y) ⊂ Ej (x),
j = 1, 2, · · · , m,
y∈Vertex(X,σ)
and
Ej (x) ⊂
(2)
[
Ej (y),
y∈Vertex(X,σ)
j = 1, 2, · · · , m.
For any simplex ∆ of (X, τ ) (after subdivision),
(3)
#(
\
x∈Vertex(∆)
Ej (x)) ≥ Kj −
n(n + 1)
,
2
j = 1, 2, · · · , m,
where n = dim X.
(When we apply this lemma in §4, the simplex X will be a simplex of a simplicial
complex Y , and Kj >> (dim Y )3 ; from this it follows that
#(
\
x∈Vertex(∆)
Ej (x)) ≥ Kj −
n(n + 1)
>> (dim Y )3 ,
2
j = 1, 2, · · · , m.)
S
Remark 3.5. The inclusion Ej (x) ⊂ y∈Vertex(X,σ) Ej (y) in the condition
(2) of Lemma 3.4 is important for our application in §4 (see 1.5.19 for the
explanation). We will put this inclusion into a more general context, 3.7. So
we will only discuss
the condition (1) in this remark.
T
The inclusion y∈Vertex(X,σ) Ej (y) ⊂ Ej (x) in the condition (1) above, will
not be used in our application in §4. But taking this inclusion as a part of the
conclusion will make the induction argument easier in the proof.
We would like to point out that the weak version of the above lemma without requiring the inclusion in (1) will automatically imply the above stronger
version. (This can be seen from the proof of Corollary 3.14 below.)
Documenta Mathematica 7 (2002) 255–461
336
Guihua Gong
3.6. Strategy and logistics of the proof of Lemma 3.4
We shall prove it by induction. So we assume that the lemma is true for simplex
of dimension at most n − 1, and prove the case that dim(X) = n.
First, we introduce a new vertex, which is the barycenter of X, and introduce
model
of E for the new vertex.
a model grouping E1model , E2model , · · · , Em
We shall view X as many layers similar to the boundary ∂X: ∂X × {0}, ∂X ×
{t1 }, · · · , and the top layer ∂X × {1} is identified into a single point which
is the barycenter, where t1 , t2 , · · · , is a finite sequence of increasing numbers
between 0 and 1 (the number of terms in this sequence depending in a certain
sense on the distance between the giving groupings at the vertices and the
model grouping at the barycenter). See the picture below.
✁✁❆
✁ ❆
❆
✁
❆
✁
❆
✁
❆
✁
✁❆
❆
✁
✁ ❆
❆
✁
✁
❆
❆
✁
✁
❆
❆
✁
✁
❆
❆
✁
✁
✁❆
❆
❆
✁
✁
✁
❆
❆
❆
✁
✁
✁
❆
❆
❆
✁
✁
✁
❆
❆
❆
✁
✁
✁
❆
❆
❆
✁
✁
✁
❆
❆
❆
✁
✁♣ ♣
✁
❆
❆
❆
✁❆
♣
✁
✁
✁
❆
❆
✁ ①❆
❆
✁
✁
✁
❆
❆
✁
❆
❆
✁
✁
✁
❆
❆
❆
✁
✁
✁
❆
❆
❆
✁
✁
✁
❆
❆
❆
✁
❆
✁
✁
❆
❆
✁
∂X
×
{t
}
✁
2
❆
❆
✁
❆
✁
❆
✁
∂X × {t1 }
❆
✁
❆❆
✁
∂X × {0}
We will introduce a subdivision of each layer ∂X × {ti } (identifying the layer as
a set with ∂X and thereby endow it with a simplicial complex structure), and
a grouping for each vertex on this layer. The general principle we shall follow
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
337
is: the higher the layer is, the closer the groupings are to the model grouping.
We should, gradually, change the groupings from each layer to the next higher
layer.
Let us explain it for the case dim(X) = 2 and dim(∂X) = 1. Fix a ti , and
suppose that we have the simplicial structure and groupings for all vertices on
∂X × {ti }. Let us using the following picture to show the vertices of ∂X × {ti }.
✉x4
✁❆
✁ ❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
❆
x5 ✉
✁
❆
✁
❆
✁
❆
✁
❆
✁
❆
∂X × {ti } ✁
❆
✁
❆
✁
❆
✁
❆
❆✉
✁
✉
✉
x1
x2
x3
(I.e., there are five 1-dimensional simplices [x1 , x2 ], [x2 , x3 ], [x3 , x4 ], [x4 , x5 ] and
[x5 , x1 ].)
Let us assume that the condition (3) holds for any simplex of ∂X × {ti } with
dim(X) replaced by dim(∂X). (We will also discuss the condition (1) below,
but not the condition (2).)
We shall construct simplicial structure and groupings on ∂X ×{ti+1 }. To begin
with, let us provisionally define the simplicial structure on ∂X × {ti+1 } to be
as the same as that on ∂X × {ti }, as in the picture on the next page.
T5
Fix an element λ ∈ E1model such that λ ∈
/ k=1 E1 (xk ). (If such an element
does not exist, then the groupings are already good for E1 . In other words,
E1 (xk ) contains and therefore equals E1model for every k. Then we should go
on to E2 or other parts.)
The grouping on the vertex yj , j = 1, · · · , 5 will be taken to be either the
grouping on the corresponding vertex xj , if E1 (xj ) ∋ λ, or the grouping on the
corresponding vertex xj , with a certain element of E1 (xj )\E1model replaced by
λ ∈ E1model if E1 (xj ) 6∋ λ. Lemma 3.9 below tells which element should be chosen to be replaced. Of course the other part Et , t > 1 of the grouping must also
Documenta Mathematica 7 (2002) 255–461
338
Guihua Gong
be slightly modified. Lemma 3.9 also guarantees that such modification exists.
Subsections 3.7 and 3.8 give the definition used in 3.9. (This consideration are
all in order to ensure the condition (2).)
①x4
✁❆
✁
✁ ❆
❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
y4
❆
✁
①
❆
✁
✁❆
❆
✁
✁
❆
✁ ❆
✁
❆
❆
✁
✁
❆
❆
✁
x5 ✁①
❆
❆
✁
✁
❆
❆
✁
✁
❆
❆
y
①
✁
5
✁
❆
❆
✁
✁
❆
❆
✁
✁
❆
❆
✁
✁
❆
❆
✁
✁
❆
❆
✁
∂X × {ti } ✁
∂X × {ti+1 }
❆
❆
✁
✁
❆
❆
①
①
❆
①
✁
✁
❆
✁
y1
y2
y3
❆
✁
❆
✁
❆❆①
①
①
✁
x1
x2
x3
T5
T5
Now, k=1 E1 (yk ) contains one more element of E1model than k=1 E1 (xk ),
namely, λ. So for E1 , the groupings on ∂X × {ti+1 } are (globally) closer to the
model grouping than that on ∂X × {ti }.
But the groupings on ∂X×{ti+1 } may not satisfy the condition (3) with dim(X)
replaced by dim(∂X), as the groupings on ∂X × {ti } do .
By the induction assumption, applied to each individual simplex of ∂X ×{t i+1 }
(with the provisional simplicial structure), we can introduce a subdivision for
∂X × {ti+1 } and groupings for the new vertices to make the condition (3), with
dim(X) replaced by dim(∂X), hold for ∂X × {ti+1 }. The picture now looks
like
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
339
①x4
✁✁❆
✁ ❆
❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
❆
✁
y4
❆
✁
①
❆
✁
❆
✁✁❆
✁
❆
❆
✁
✁
❆
❆
✁
✁
❆
❆
✁
①
′′
x5 ✁
❆①y3
❆
✁
✁
❆
❆
✁
✁
❆
❆
y5 ✁①
✁
❆
❆
✁
✁
❆ ①y ′
❆
✁
✁
3
❆
❆
✁
′
✁
y5 ①
❆
❆
✁
✁
❆
❆
✁
∂X × {ti } ✁
❆
❆
✁ ∂X × {ti+1 }
✁
❆
❆
❆
✁
①
①
①
①
①
✁
❆
′
′
✁
y3
y2
y2
y1
y1
❆
✁
❆
✁
❆❆①
✁
①
①
x1
x2
x3
In this picture, y1′ , y2′ , y3′ , y3′′ , and y5′ are the new vertices introduced in the
subdivision.
(Of course this picture only shows a special case.)
It goes without saying that we wish to ensure the condition (1) (and also
the condition (2)) for the groupings associated to the new vertices inside each
provisional simplex of ∂X × {ti+1 }. In the other words, when we introduce the
groupings for a new vertex inside a fixed provisional simplex of ∂X × {ti+1 }
(e.g., y2′ inside [y2 , y3 ]), for each k we should keep the intersection of the sets
Ek over vertices of this simplex (e.g., Ek (y2 ) ∩ Ek (y3 )) inside the set Ek for the
new vertex (e.g., inside Ek (y2′ )). This is the condition (1) for this provisional
simplex. The condition (1) for all the individual simplices
implies that after the
T
subdivision, the intersection over the whole layer y∈Vertex(X×{ti+1 }) E1 (y) is
equal
to the intersection over the vertices of provisional simplicial structure
T5
model
E
than
1 (yk ), and therefore still contains one more element of E1
Tk=1
5
E
(x
)
(namely
,
λ).
k=1 1 k
Documenta Mathematica 7 (2002) 255–461
340
Guihua Gong
One may notice that the subset ∂X × [ti , ti+1 ] is not automatically a simplicial
complex. We shall use Lemma 3.10 below to decompose it into a simplicial
complex.
Because we do not change much from the grouping of xj to the grouping of
yj and because we make (1) true when introduce groupings for new vertices
yj′ , yj′′ , etc., the groupings for any simplex inside ∂X × [ti , ti+1 ] will satisfy the
condition (3) (of course with dim(X) not replaced by dim(∂X)).
Finally, let us mention that, we carry out the above construction separately for
E1 , E2 , etc. Once this has been done for E1 , the same method can be used for
E2 . The condition (1) will guarantee that when we work on E2 , we will not
affect the condition (3) for E1 , which was supposed to be already satisfied.
(The details will be contained in the proof of Lemma 3.11)
As we mentioned above, when we construct E1 (yj ) from E1 (xj ), we need to
replace one element of E1 (xj )\E1model by the element λ ∈ E1model \E1 (xj ). If we
choose an arbitrary element µ ∈ E1 (xj )\E1model to be replaced by λ to define
E1 (yj ), then in general, E1 (yj ) may not be extended to a grouping satisfying
the condition (2), in other words, there may not exist a grouping E1 , E2 , · · · , Em
of E of type (K1 , K2 , · · · , Km ) such that E1 = E1 (yj ) and
Ek ⊂
[
x∈Vertex(X)
Ek (x), k = 1, 2, · · · , m.
So we need to give a condition to ensure that a subset E1 ⊂ E can be extended
to a grouping satisfying condition (2). This will be discussed in 3.7 and 3.8.
(See condition (∗∗) in 3.8.)
The proof of Lemma 3.4 will be given in 3.7 to 3.16.
S
3.7. We will put the inclusion Ej (x) ⊂ y∈Vertex(X,σ) Ej (y) in the condition
(2) of Lemma 3.4, into a more general form, as follows. (In fact, we will use
this more general form in our application.)
Suppose that H1 , H2 , · · · , Hm are (not necessarily disjoint) subsets of E, satisfying the following condition (called Condition (∗)). For each subset I ⊂
{1, 2, · · · , m},
#(
[
i∈I
Hi ) ≥
X
Ki .
(∗)
i∈I
It follows obviously that H1 ∪H2 ∪· · ·∪Hm = E, since #(H1 ∪H2 ∪· · ·∪Hm ) ≥
P
m
i=1 Ki = #(E).
From the Marriage Lemma of [HV] (or the Pairing Lemma in [Su]), the condition (∗) is a necessary and sufficient condition for the existence of a grouping
E1 , E2 , · · · , Em of E of type (K1 , K2 , · · · , Km ) with the condition Ei ⊂ Hi .
(Recall that, the Marriage Lemma of [HV] is stated as follows.
Suppose that there are two groups of K boys and K girls. Suppose that the
following condition holds:
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
341
For any subset of K1 girls (K1 = 1, 2, · · · , K), there are at least K1 boys, each
of them knows at least one girl from this subset.
Then there is a way to arrange marriage between them such that each boy
marries one of the girls he knows.
Our claim above is a special case of this Marriage Lemma. One can see this as
follows. Suppose that the K girls are from m different clubs, and the i-th club
has exactly Ki girls. Number the boys by 1, 2, · · · , K. Let us define the relation
consisting of a boy knowing a girl as follows. If j ∈ Hi , then the boy j knows
all the girls in the i-th club. Otherwise, he does not know any girl in the i-th
club. (Notice that the boy j could be in different Hi , so he could know girls
from different clubs.) Obviously the condition (*) becomes the above condition
in the Marriage Lemma. So if the condition (∗) holds, then there is a way to
arrange the marriage as in the lemma. One can define Ei to be the set of boys
each of whom marries a girl from the i-th club. Obviously Ei ⊂ Hi . This
proves the sufficiency
part of the condition. The necessary part is trivial.)
S
If we let Hi = y∈Vertex(X,σ) Ei (y), then the inclusion in (2) of Lemma 3.4
becomes Ei (x) ⊂ Hi for each x ∈ Vertex(X, σ).
For any subset I ⊂ {1, 2, · · · , m}, let
[
HI =
Hi .
i∈I
3.8. We say that a subset E1 ⊂ H1 , of K1 elements, satisfies Condition (∗∗)
if for any I ⊂ {2, 3, · · · , m},
X
(∗∗)
#(HI \E1 ) ≥
Ki .
i∈I
(Caution: 1 ∈
/ I.) Again, from the Marriage Lemma, E1 ⊂ H1 satisfies (∗∗)
if and only if E1 can be extended to a grouping E1 , E2 , · · · , Em of E of type
(K1 , K2 , · · · , Km ) such that Ei ⊂ Hi .
Lemma 3.9. Suppose that E1 , F1 (⊂ H1 ) are two subsets satisfying (∗∗). If
λ ∈ F1 \E1 , then there is a µ ∈ E1 \F1 such that
E1′ = (E1 \{µ}) ∪ {λ},
satisfies (∗∗).
Proof: Let G = E1 ∪ {λ}. Since E1 satisfies (∗∗), necessarily,
X
#(HI \G) ≥
Ki − 1 ,
i∈I
for all subsets I ⊂ {2, 3, · · · , m}.
Let H̃i = Hi \G, i ∈ {2, 3, · · · , m}. And let H̃I =P∪i∈I H̃i for any I ⊂
{2, 3, · · · , m}. The above inequality becomes #(H̃I ) ≥ i∈I Ki − 1 .
Documenta Mathematica 7 (2002) 255–461
342
Guihua Gong
Let I0 be a minimum subset of {2, 3, · · · , m} such that
X
Ki − 1 .
#(H̃I0 ) =
i∈I0
Note that such set I0 exists, since if I = {2, 3, · · · , m},
X
#(H̃I ) =
Ki − 1 .
i∈I
Using the fact that #(HI0 \F1 ) ≥
P
i∈I0
Ki , we can prove that
E1 ∩ HI0 6⊂ F1 .
If it is not true, then G ∩ HI0 ⊂ F1 , since λ ∈ F1 and G = E1 ∪ {λ}. And
therefore,
X
Ki ,
#(H̃I0 ) = #(HI0 \G) ≥ #(HI0 \F1 ) ≥
i∈I0
which contradicts with the above equation.
Choose any element µ ∈ (E1 ∩ HI0 )\F1 ; we will prove that µ is as desired in
the lemma. I.e., the set
E1′ = (E1 \{µ}) ∪ {λ} = G\{µ}
satisfies (∗∗). That is, for any J ⊂ {2, 3, · · · , m}, #(HJ \E1′ ) ≥
The proof is divided into three cases.
(i) The case that J ∩ I0 = ∅. By the relations
H̃I0 ∪J = (H̃J \H̃I0 ) ∪ H̃I0
and
#(H̃I0 ∪J ) ≥
P
i∈J
(disjoint union)
X
i∈I0 ∪J
Ki − 1 ,
combined with the definition of I0 , one knows that
X
(a)
#(H̃J \H̃I0 ) ≥
Ki ,
i∈J
which is stronger than the condition
#(HJ \E1′ ) ≥
X
Ki .
i∈J
(ii) The case that J ⊂ I0 . Obviously, for J = I0 , we have
X
Ki ,
#(HI0 \E1′ ) = #(HI0 \G) + 1 =
i∈I0
Documenta Mathematica 7 (2002) 255–461
Ki .
Simple Inductive Limit C ∗ -Algebras, I
343
since E1′ = G\{µ} and µ ∈ HI0 ∩ G.
So we can suppose that J ⊂
6= I0 .
By the minimality of I0 , we know that
#(HJ \G) ≥
X
Ki .
i∈J
Therefore,
#(HJ \E1′ ) ≥
X
Ki .
i∈J
(iii) The general case. Let J0 = J ∩ I0 , and J1 = J\J0 . Then
(HJ \E1′ ) ⊃ (H̃J1 \H̃I0 ) ∪ (HJ0 \E1′ ),
where the right hand side is a disjoint union since J0 ⊂ I0 .
Evidently, this case follows from (a) above and case (ii).
The following lemma is perhaps well known.
⊔
⊓
Lemma 3.10. Let (X, σ0 ) be a simplicial complex and (X, σ1 ) be a subdivision
of (X, σ0 ). It follows that there is a simplicial structure σ of X × [0, 1] such
that
(1) all vertices of (X × [0, 1], σ) are on three subsets X × {0}, X × { 21 }, and
X × {1};
(2) (X × [0, 1], σ)|X×{0} = (X, σ0 ), and (X × [0, 1], σ)|X×{1} = (X, σ1 );
(3) For a simplex ∆ of (X ×[0, 1], σ), there is a simplex ∆0 of (X, σ0 ) (caution:
we do not use (X, σ1 )) such that
∆ ⊂ ∆0 × [0, 1],
as a subset.
Proof: We prove it by induction on dim(X).
If X is 0-dimensional simplicial complex which consists of finitely many points,
the conclusion is obvious, since X × [0, 1] is finitely many disjoint intervals.
(Note that, at this case, necessarily, (X, σ0 ) = (X, σ1 ).) For us to visualize
the general case later on, we introduce a new vertex (x, 12 ) ∈ X × { 12 } for each
x ∈ X. That is, we divide the interval {x} × [0, 1] into two simplices {x} × [0, 12 ]
and {x} × [ 12 , 1].
As the induction assumption, let us assume that the lemma is true for any
n-dimensional complex. Let dim(X) = n + 1.
Let X (n) be the n-skeleton of (X, σ0 ) (we use σ0 not σ1 here). By the induction
assumption, there is a simplicial structure σ ′ of X (n) × [0, 1] such that
(1) all vertices of (X (n) × [0, 1], σ ′ ) are on three subsets X (n) × {0}, X (n) × { 12 },
and X (n) × {1};
Documenta Mathematica 7 (2002) 255–461
344
Guihua Gong
(2) (X (n) × [0, 1], σ ′ )|X (n) ×{0} = (X (n) , σ0 |X (n) ), and
(X (n) × [0, 1], σ ′ )|X (n) ×{1} = (X (n) , σ1 |X (n) );
(3) For an simplex ∆ of (X (n) × [0, 1], σ ′ ), there is a simplex ∆0 of (X (n) , σ0 )
such that
∆ ⊂ ∆0 × [0, 1],
as a subset.
Let us introduce the simplicial structure σ on X × [0, 1] such that
(X × [0, 1], σ)|X (n) ×[0,1] = (X (n) × [0, 1], σ ′ ), (X × [0, 1], σ)|X×{0} = (X, σ0 ),
and (X × [0, 1], σ)|X×{1} = (X, σ1 ).
Consider each ∆ × [0, 1] for any (n + 1)-simplex ∆ of (X, σ0 ) (again, we use
σ0 not σ1 ). From the above, we already have the simplicial structure on the
boundary
∂(∆ × [0, 1]) = (∆ × {0}) ∪ (∂∆ × [0, 1]) ∪ (∆ × {1}).
Namely, on ∆ × {0}, we use σ0 ; on ∂∆ × [0, 1], we use σ ′ ; and on ∆ × {1}, we
use σ1 .
Let c be the barycenter of ∆, introduce a new vertex C = (c, 21 ) ∈ X × { 21 }.
The simplices of σ on ∆ × [0, 1] are of the following forms.
(i) C itself is a zero dimensional simplex;
(ii) Any simplex of the boundary ∂(∆ × [0, 1]) is a simplex for σ on ∆ × [0, 1];
and
(iii) For any simplex ∆′ of the boundary ∂(∆×[0, 1]), the convex hull of ∆′ ∪{C}
is a simplex of dimension (dim(∆′ ) + 1) for σ on ∆ × [0, 1].
Define such simplicial structure for each (n + 1)-simplex separately, and put
them together give rise to a simplicial structure of X × [0, 1], which obviously
satisfies the conditions (1), (2), and (3).
(Note that the simplicial structure on ∂∆ × [0, 1] is as the same as σ ′ , therefore
the simplicial structure on ∆ × [0, 1] and on ∆1 × [0, 1] for different (n + 1)dimensional simplices ∆ and ∆1 are compatible on the intersection (∆ ∩ ∆1 ) ×
[0, 1].)
The following pictures may help the reader to visualize the construction. They are pictures only for the case n = 0, dim(∆) = 1, and
dim(∆ × [0, 1]) = 2.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
345
Suppose the simplicial structure for the boundary ∂(∆ × [0, 1]) is as follows.
(The dots represent vertices.)
①
①
①
①
①
①
①
①
∆ × {0}
∆ × {1}
Then the simplicial structure on ∆ × [0, 1] will be described by the following
picture.
①
◗
◗
◗
◗
①
◗
◗
◗
◗
◗
◗
◗
◗
✑
✑
✑
✑
✑
✑
✑
✑
✑
✑
①
✑
✑
✥①
✥✥
✑
✑
✥✥
✥
✥
✥
◗
✥
◗
✑ ✥✥
✑
✥✥
✥✥
◗
◗❵
✑✥
✥
✥
①
✑
❵
❵❵
◗❵
❵❵
✑◗
✑
❵❵
❵❵
◗
❵❵
◗
✑C
✑
❵❵
❵❵
❵①
◗
◗
✑
✑
◗
◗
✑
✑
◗
◗
✑
✑
◗
◗
✑
✑
◗
◗
✑
✑
◗
◗
✑
✑
◗
◗①
✑
✑
①
①
∆ × {0}
◗
◗
∆ × { 12 }
∆ × {1}
⊔
⊓
Documenta Mathematica 7 (2002) 255–461
346
Guihua Gong
The following lemma presents the main technical step of this section.
Lemma 3.11. Suppose that {H1 , H2 , · · · , Hm } satisfies the condition (∗). Suppose that (X, σ) is a simplicial complex consisting of a single simplex ∆0 and
its faces. Let (Y, σ) = (∂∆0 , σ), and (Y, τ ) be a subdivision of (Y, σ). Suppose
that it is assigned, for each vertex x ∈ (Y, τ ), a set E1 (x) ⊂ H1 which satisfies
the condition (∗∗). Furthermore, suppose that for any simplex ∆ of (Y, τ ),
\
dim Y (dim Y + 1)
.
#
E1 (y) ≥ K1 −
2
y∈Vertex(∆)
It follows that there are a subdivision (X, τ̃ ) of (X, σ) and an assignment, for
each vertex x ∈ Vertex(X, τ̃ ), a set E1 (x) ⊂ H1 , satisfying condition (∗∗), with
the following conditions.
(1) (X, τ̃ )|Y = (Y, τ ), and for each vertex y ∈ Vertex(Y, τ ), the assignment
E1 (y) is as same as the original one.
(2) For any x ∈ (X, τ̃ ),
\
E1 (x) ⊃
E1 (y) .
y∈Vertex(Y,τ )
(3) For any simplex ∆ of (X, τ̃ ),
\
dim X(dim X + 1)
#
E1 (x) ≥ K1 −
.
2
x∈Vertex(∆)
Proof: The Lemma is proved by induction on the dimension of the simplex.
If dim(∆0 ) = 0, then ∆0 = {pt}, a set of single point, and ∂∆0 = ∅. Obviously,
the lemma holds by choosing any E1 (pt) ⊆ H1 of K1 element to satisfy (∗∗).
Let us prove the 1-dimensional case. Logically, this part could be skipped. But
the proof of this case will be easier to visualize which can be used to understand
the general case.
Suppose that dim(∆0 ) = 1. ∆0 is a line segment [0, 1]. Divide [0, 1] into several
subintervals by
0 = t00 < t01 < t02 < · · · < t0a−1 < t0a =
1
= t1a < t1a−1 < · · · < t12 < t11 < t10 = 1.
2
(The natural number a is to be determined later.) The points {tij }i=0,1;j=1,2···,a
will be the new vertices of (∆0 , τ̃ ). (Note that t0a is the same vertex as t1a .)
Choose a model E1model ⊂ H1 to satisfy (∗∗) and
E1model ⊃ E1 (t00 ) ∩ E1 (t10 ).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
347
In fact, one can choose E1model to be either E1 (t00 ) or E0 (t10 ). (Note that t00 = 0
and t10 = 1 are vertices of (∆0 , σ).)
Let G = E1 (t00 ) ∩ E1 (t10 ) ∩ E1model . Without loss of generality, we can assume
that there is λ ∈ E1model \G. Otherwise, E1model = G = E1 (t00 ) = E1 (t10 ), and
the conclusion already holds before introducing any subdivision.
By Lemma 3.9, if λ ∈
/ E1 (ti0 ) (i = 0, 1), then there is a µ ∈ E1 (ti0 )\E1model such
i
that E1 (t0 ) ∪ {λ}\{µ} satisfies (∗∗). Define
½
if λ ∈ E1 (ti0 )
E1 (ti0 )
E1 (ti1 ) =
i
(E1 (t0 ) ∪ {λ})\{µ} if λ ∈
/ E1 (ti0 ) .
Then E1 (ti1 ) ⊃ G ∪ {λ}. Therefore,
E1 (t01 ) ∩ E1 (t11 ) ∩ E1model ⊃
E (t0 ) ∩ E1 (t10 ) ∩ E1model .
6= 1 0
Suppose that we already have the definitions of E1 (t0i ) and E1 (t1i ), we can
define E1 (t0i+1 ) and E1 (t1i+1 ) exactly the same as above (i in place of 0, and
i + 1 in place of 1), and obtain
E1 (t0i+1 ) ∩ E1 (t1i+1 ) ∩ E1model ⊃
E (t0 ) ∩ E1 (t1i ) ∩ E1model .
6= 1 i
Carrying out this procedure for at most finitely many times, we will reach
E1 (t0a−1 ) ∩ E1 (t1a−1 ) ∩ E1model = E1model . Then define E1 (tia ) = E1model . (Note
that t1a = t0a = 12 .)
For i = 0, 1; j = 0, 1, 2, · · · , a − 1,
#(E1 (tij ) ∩ E1 (tij+1 )) ≥ K1 − 1 = K1 −
dim(∆0 )(dim(∆0 ) + 1)
,
2
since we take out at most one point from E1 (tij ) to define E1 (tij+1 ). This proves
that the lemma holds for n = 1.
(Let us point out that for one dimensional case, the proof could be simpler.
We choose the above proof to present some idea for the general case below.)
Suppose that the lemma is true for any simplex of dimension ≤ n − 1. We will
prove it for dim(X) = n. (One should compare to the explanation in 3.6.)
Step 1. Identify ∆0 with ∂∆0 × [0, 1]/∂∆0 × {1}. Regard ∂∆0 as ∂∆0 × {0} ⊂
∂∆0 × [0, 1]. Note that ∂∆0 × {1} is identified as a single point which is the
center of ∆0 , and is NOT a vertex of (∆0 , σ).
Choose 0 = t0 < t1 < · · · < ta = 1. (The natural number a is to be determined
later.)
We will first introduce some new vertices (for the subdivision (X, τ̃ )) on ∂∆0 ×
{t1 }, ∂∆0 × {t2 }, · · · , ∂∆0 × {ta }, and define E1 for those vertices.
Later on (in Step 4), we will consider each ∂∆ × [ti , ti+1 ] to be X × [0, 1] in
Lemma 3.10, and introduce new vertices on ∂∆×{ ti +t2i+1 } (in place of X ×{ 21 }).
(We need to do this, because ∂∆ × [ti , ti+1 ] is not automatically a simplicial
complex.)
Documenta Mathematica 7 (2002) 255–461
348
Guihua Gong
Choose a model E1model ⊂ H1 to satisfy (∗∗). We also require that
\
E1model ⊃
E1 (y) .
y∈Vertex(Y,τ )
(One can choose E1model to be E1 (y) for any vertex y ∈ Vertex(Y, τ ).)
Define E1 {(x, 1)} = E1model . (Note that {(x, 1)} ⊂ ∂∆0 × [0, 1] is identified to
a single point, the center of ∆0 .)
The construction will be carried out in Step 2, 3 and 4. The procedure can be
outlined as follows. If we already have the construction of simplicial structure τ̃
for ∂∆0 ×{ti−1 } and the definition of E1 on all vertices in Vertex(∂∆0 ×{ti−1 }),
then, to define the simplicial structure on ∂∆0 × [ti−1 , ti ] (in particular, to
introduce vertices on ∂∆0 × {ti }), and to define E1 on the newly introduced
vertices (on ∂∆0 × {ti }), we will only use the simplicial structure and the
definition of E1 on ∂∆0 × {ti−1 }.
In this procedure, if there is a vertex x ∈ Vertex(∂∆0 × {ti−1 }, τ̃ ) such that
E1 (x) 6= E1model ,
then we will require that
\
(a)
E1 (x) ∩ E1model ⊃
6=
x∈Vertex(∂∆0 ×{ti },τ̃ )
\
E1 (x) ∩ E1model .
x∈Vertex(∂∆0 ×{ti−1 },τ̃ )
(That is, the sets E1 ’s on ∂∆0 × {ti } are globally closer to E1model than those
on ∂∆0 × {ti−1 }.) Finally, within finitely many steps, we will reach that, for
certain i − 1, and for all vertices x ∈ Vertex(∂∆0 × {ti−1 }, τ̃ ),
E1 (x) = E1model .
Then we choose ti = ta = 1, and choose any simplicial structure on ∂∆0 ×
[ta−1 , 1]/∂∆0 × {1} with vertex set to be Vertex(∂∆0 × {ta−1 }) ∪ ∂∆0 × {1}.
Recall that the set ∂∆0 ×{1} is identified as a single point with E1 (∂∆0 ×{1}) =
E1model .)
Furthermore, in this procedure, we not only make (3) true for any simplex in
∂∆0 × [ti−1 , ti ], but also make the following stronger statement true for any
simplex ∆ lies on ∂∆0 × {ti }:
\
(n − 1)n
.
(b)
#
E1 (y) ≥ K1 −
2
y∈Vertex(∆)
(Note that n − 1 = dim(∂∆0 × {ti }) = dim(∂∆0 ).) This condition has to be
satisfied for the construction of the next step by induction.
Step 2. We will do all the above construction only for ∂∆0 × [t0 , t1 ]. For the
other part of the construction, one uses induction argument with aid of (b)
(i.e., let ti−1 play the role of t0 , and ti play the role of t1 .)
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
349
Let {(y1 , t0 ), (y2 , t0 ), · · · , (yp , t0 )} be the vertices of ∂∆0 × {t0 } = Y . There is a
simplicial complex structure on ∂∆0 ×{t1 }, which is exactly the same as that of
(∂∆0 × {t0 }, τ̃ ), since both ∂∆0 × {t1 } and ∂∆0 × {t0 } can be regarded as ∂∆0 .
We call such simplicial complex τ̃pre . Therefore, each point (yi , t1 ) (1 ≤ i ≤ p)
is a vertex of (∂∆0 × {t1 }, τ̃pre ). We will introduce more vertices later.
Let G = E1 (y1 , t0 ) ∩ E1 (y2 , t0 ) ∩ · · · ∩ E1 (yp , t0 ) ∩ E1model . If G = E1model , then
E1 (yi , t0 ) = E1model for each 1 ≤ i ≤ p and the construction is done. So we
assume
G 6= E1model .
Choose λ ∈ E1model \G. When we define E1 (x) for any vertex x ∈ ∂∆0 × {t1 },
it is always required that
E1 (x) ⊃ G ∪ {λ}.
Therefore, (a) holds for the pair {t0 , t1 }.
For each point (yi , t0 ), if λ ∈
/ E1 (yi , t0 ), by Lemma 3.9, there is µ ∈
E1 (yi , t0 )\E1model such that (E1 (yi , t0 ) ∪ {λ})\{µ} satisfies (∗∗). Define
E1 (yi , t1 ) =
½
E1 (yi , t0 )
(E1 (yi , t0 ) ∪ {λ})\{µ}
if λ ∈ E1 (yi , t0 )
if λ ∈
/ E1 (yi , t0 ) .
In this way, obviously, E1 (x) ⊃ G ∪ {λ} for each vertex x = (yi , t1 ) ∈
Vertex(∂∆0 × {t1 }, τ̃pre ).
Step 3. Note that the definition of E1 on Vertex(∂∆0 × {t1 }, τ̃pre ) may not
satisfies (b). Therefore we can not use the simplicial structure τ̃pre and the
definition of E1 on Vertex(∂∆0 × {t1 }, τ̃pre ) to construct simplicial structure
and the definition of E1 for ∂∆0 × {t2 }. We need to introduce a subdivision
for (∂∆0 × {t1 }, τ̃pre ) and the definitions of E1 for new vertices to make (b)
true. (This step is not needed in the one dimensional case, since for any zero
dimensional simplex (which is a point), (b) automatically holds.)
Apply the induction assumption to each simplex of (∂∆0 × {t1 }, τ̃pre ) with the
above definition of E1 on Vertex(∂∆0 × {t1 }, τ̃pre ), from the simplices of the
lowest dimension ( dimension 1) to the simplices of the highest dimension (dimension n − 1). (Note that each such simplex has dimension at most n − 1.)
One should begin with each 1-simplex (with boundary being two points — two
0-simplices), then each 2-simplex, and so on.
First, let e be any 1-simplex of (∂∆0 × {t1 }, τ̃pre ) with boundary ∂e = {v0 , v1 }.
Obviously, the condition of Lemma 3.11 automatically holds for simplex e in
place of ∆0 and ∂e in place of ∂∆0 , since ∂e is zero-dimensional. By the
induction assumption, there is a subdivision (e, τ̃ ) of (e, τ̃pre ) and the definition
of E1 for each vertex of (e, τ̃ ) such that
(1) The definition of E1 on the original vertices {v0 , v1 } are the same as before.
(2) For any x ∈ Vertex(e, τ̃ ),
E1 (x) ⊃ E1 (v0 ) ∩ E1 (v1 ).
Documenta Mathematica 7 (2002) 255–461
350
Guihua Gong
(3) For any simplex e′ of (e, τ̃ ) (a line segment of e)
\
x∈Vertex(e′ )
E1 (x) ≥ K1 −
dim(e)(dim(e) + 1)
.
2
After we have done the above procedure for each 1-simplex, we can do it for
each 2-simplex, since we already have simplicial structure and the definition of
E1 for the boundary of any 2-simplex as required in the condition of Lemma
3.11.
Going through this way, finally, one obtains a subdivision (∂∆0 × {t1 }, τ̃ ) of
(∂∆0 × {t1 }, τ̃pre ) and the definition of E1 for each newly introduced vertex,
such that the following two statements hold.
1. For each old simplex ∆ of (∂∆0 × {t1 }, τ̃pre ) and any new vertex x ∈ ∆,
E1 (x) ⊃
(c)
\
E1 (y) .
y∈Vertex(∆,τ̃pre )
2. If ∆ is a simplex of (∂∆0 × {t1 }, τ̃ ), then
#
\
y∈Vertex(∆)
E1 (y)
≥ K1 −
(n − 1)n
dim Y (dim Y + 1)
= K1 −
.
2
2
(This is the requirement (b) in Step 1.)
The first statement is the induction assumption of validity of (2) and the second
statement is the induction assumption of validity of (3).
Step 4. In this step, we will apply Lemma 3.10 to define the simplicial structure τ̃ on ∂∆0 × [t0 , t1 ] and the definitions of E1 on all vertices. Note that we
already have simplicial structure τ̃ on ∂∆0 × {t0 } and on ∂∆0 × {t1 }. Furthermore, τ̃ |∂∆0 ×{t1 } is a subdivision of τ̃ |∂∆0 ×{t0 } if we regard both ∂∆0 × {t0 }
and ∂∆0 × {t1 } as ∂∆0 . Apply Lemma 3.10 (with ∂∆0 in place of X) to obtain
the simplicial structure on ∂∆0 ×[t0 , t1 ] (we only need to introduce new vertices
1
on ∂∆0 × { t0 +t
2 }).
t0 +t1
1
For each new vertex (u, t0 +t
2 ) ∈ ∂∆0 × { 2 }, consider (u, t0 ) ∈ ∂∆0 × {t0 }.
From 3.1, there is a unique simplex ∆ of (∂∆0 × {t0 }, τ̃ ) such that (u, t0 ) ∈
1
interior(∆). Choose any vertex x of ∆ and define E1 (u, t0 +t
2 ) = E1 (x).
So we have the simplicial structure τ̃ on ∂∆0 × [t0 , t1 ] and the definition of
E1 (x) for each x ∈ Vertex(∂∆0 × [t0 , t1 ]). We need to verify the condition (3).
Let ∆ be any simplex of (∂∆0 × {t0 }, τ̃ ) with vertices {(u0 , t0 ), (u1 , t0 ), · · · ,
(ui , t0 )}. Then
#(E1 (u0 , t0 ) ∩ E1 (u1 , t0 ) ∩ · · · ∩ E1 (ui , t0 )) ≥ K1 −
dim Y (dim Y + 1)
.
2
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
351
Let G1 = E1 (u0 , t0 ) ∩ E1 (u1 , t0 ) ∩ · · · ∩ E1 (ui , t0 ). From the above definition of
t0 +t1
1
E1 for vertices of ∂∆0 × { t0 +t
2 }, we know that if (u, t0 ) ∈ ∆ and (u,
2 ) ∈
Vertex(∂∆0 × [0, 1], τ̃ ), then
(d)
E1 (u,
t0 + t 1
) ⊃ G1 .
2
Since each E1 (uj , t1 ) is either E1 (uj , t0 ) or is obtained by replacing one element
of E1 (uj , t0 ) by λ, we have
#(G1 ∩ E1 (u0 , t1 ) ∩ E1 (u1 , t1 ) ∩ · · · ∩ E1 (ui , t1 ))
Y +1)
≥ K1 − dim Y (dim
− (i + 1)
2
(n−1)n
≥ K1 − 2 − n
.
= K1 − n(n+1)
2
(e)
(Note that there are i + 1 (≤ n) sets of {E1 (uj , t1 )}ij=0 , and, therefore, at most
i + 1 points were taken out from G1 .)
Recall that τ̃ on ∆0 × {t1 } is the subdivision of τ̃pre . By (c) of Step 3, we have
\
\
E1 (x) ⊃
E1 (y)
x∈Vertex(∆×{t1 },τ̃ )
y∈Vertex(∆×{t1 },τ̃pre )
= E1 (u0 , t1 ) ∩ E1 (u1 , t1 ) ∩ · · · ∩ E1 (ui , t1 ).
(Note that (c) implies that the above “⊃” holds if the left hand side of “⊃”
is replaced by E1 (x) for any x ∈ Vertex(∆ × {t1 }, τ̃ ), so it also holds for the
intersection of these E1 (x). In fact, the above “⊃” can be replaced by “=”.)
Then combining it with (d), we have
\
x∈Vertex(∆×[t0 ,t1 ],τ̃ )
E1 (x) = G1 ∩ E1 (u0 , t1 ) ∩ E1 (u1 , t1 ) ∩ · · · ∩ E1 (ui , t1 )
elements by (e). Combining this fact with (3)
which has at least K1 − n(n+1)
2
of Lemma 3.10, we know that the desired condition (3) holds for any simplex
of (∂∆0 × [t0 , t1 ], τ̃ ).
Evidently, (2) holds from the construction.
Since (b) holds for ∂∆0 × {t1 }, one can continue this procedure. This ends the
proof.
⊔
⊓
Corollary 3.12. Suppose that {H1 , H2 , · · · , Hm } satisfies the condition (∗).
Suppose that (X, σ) is a simplicial complex consisting of a single simplex and
its faces. Suppose that there is assigned, for each vertex x ∈ (X, σ), a set
E1 (x) ⊂ H1 which satisfies the condition (∗∗).
It follows that there are a subdivision (X, τ ) of (X, σ) and an assignment, for
each new vertex x ∈ Vertex(X, τ ), a set E1 (x) ⊂ H1 , satisfying condition (∗∗),
Documenta Mathematica 7 (2002) 255–461
352
Guihua Gong
with the following conditions. (The definition of E1 for the old vertex should
not be changed.)
(1) For any x ∈ Vertex(X, τ ),
\
E1 (x) ⊃
E1 (y) .
y∈Vertex(X,σ)
(2) For any simplex ∆ of (X, τ ),
\
dim X(dim X + 1)
#
.
E1 (x) ≥ K1 −
2
x∈Vertex(∆)
Proof: To prove this corollary, one needs to apply Lemma 3.11 to simplices
from the lowest dimension (e.g. dimension one simplex whose boundary consists
two vertices of (X, σ)) to the highest dimension (e.g the simplex X itself with
boundary ∂X). Each time, we only work on a single simplex ∆ of (X, σ). And
when we work on ∆, we should assume that we already have the subdivision
and the definition of E1 on the boundary ∂∆ to satisfy the condition in Lemma
3.11 with dim(∂∆) in place of dim(Y ).
⊔
⊓
Corollary 3.13. Let (X, σ) be a simplicial complex consisting of a single
simplex X and all its faces. Suppose that associated to each x ∈ Vertex(X, σ),
there is a grouping E1 (x), E2 (x), · · · , Em (x) of E.
It follows that there is a subdivision (X, τ ) of (X, σ), and associated to each
new vertex x ∈ Vertex(∆, τ ), there is a grouping E1 (x), E2 (x), · · · , Em (x) of E
(for any old vertex of (∆, σ), the grouping should not be changed), such that
the following hold.
For each newly introduced vertex x ∈ Vertex(X, τ ),
[
(2)
Ej (x) ⊂
Ej (y),
j = 1, 2, · · · , m.
y∈Vertex(X,σ)
For any simplex ∆ of (X, τ ) (after subdivision),
(3)
#(
\
x∈Vertex(∆)
E1 (x)) ≥ K1 −
n(n + 1)
2
where n = dim X.
(In this corollary, we do not require the condition (1) in Lemma 3.4. This will
be done in the next corollary.)
Proof:
Set
S
y∈Vertex(X,σ)
Ej (y) := Hj ,
j = 1, 2, · · · , m.
Documenta Mathematica 7 (2002) 255–461
Then
Simple Inductive Limit C ∗ -Algebras, I
353
H1 , H2 , · · · , Hm satisfy condition (∗), and for each x ∈ Vertex(X, σ), E1 (x) ⊂
H1 satisfies (∗∗).
Applying Corollary 3.12, we obtain the subdivision (X, τ ) and the definition
of E1 (x), for each new vertex, to satisfy condition (∗∗), and (1) and (2) in the
Corollary 3.12.
For each new vertex x, since E1 (x) satisfies (∗∗), we can extend it to a grouping
E1 (x), E2 (x), · · · , Em (x) such that Ei (x) ⊂ Hi . Therefore this grouping satisfy
the condition (2) of our corollary.
The condition (3) follows from the condition (2) of Corollary 3.12. Thus the
corollary is proved.
⊔
⊓
Corollary 3.14. Let (X, σ) be a simplicial complex consisting of a single
simplex X and all its faces. Suppose that associated to each x ∈ Vertex(X, σ),
there is a grouping E1 (x), E2 (x), · · · , Em (x) of E.
It follows that there is a subdivision (X, τ ) of (X, σ), and associated to each
new vertex x ∈ Vertex(∆, τ ), there is a grouping E1 (x), E2 (x), · · · , Em (x) of E
(for any old vertex of (∆, σ), the grouping should not be changed), such that
the following hold.
For each newly introduced vertex x ∈ Vertex(X, τ ),
\
(1)
Ej (y) ⊂ Ej (x),
j = 1, 2, · · · , m,
y∈Vertex(X,σ)
and
(2)
Ej (x) ⊂
[
Ej (y),
y∈Vertex(X,σ)
j = 1, 2, · · · , m.
For any simplex ∆ of (X, τ ) (after subdivision),
(3)
#(
\
x∈Vertex(∆)
E1 (x)) ≥ K1 −
n(n + 1)
.
2
where n = dim X.
(Comparing this Corollary to Lemma 3.4, the only difference is that we require
(3) holds only for E1 in the corollary.)
Proof: The only difference between this corollary and Corollary 3.13 is that we
require condition (1)Tholds. To make (1) hold, we need to do the following. Reserve all the subsets y∈Vertex(X,σ) Ej (y), j = 1, 2, · · · , m, which are supposed
to be in Ej (x) (if we want the condition (1) to hold), for any newly introduced
vertex
T x; group the rest of the elements of E (using Corollary 3.13); and finally
put y∈Vertex(X,σ) Ej (y) into each Ej (x). The details are as follows.
T
Set y∈Vertex(X,σ) Ej (y) := Dj , j = 1, 2, · · · , m. Then Dj , j = 1, 2, · · · , m
are mutually disjoint. To see this, we fix a y ∈ Vertex(X, σ), and notice
Documenta Mathematica 7 (2002) 255–461
354
Guihua Gong
that Dj ⊂ Ej (y), and Ej (y), j = 1, 2, · · · , m are mutually disjoint, from the
definition of grouping. Similarly, if j1 6= j2 , then Dj1 ∩ Ej2 (y) = ∅, for any
y ∈ Vertex(X, σ).
Consider E ′ = E\(∪j Dj ) and the m-tuple
′
(K1′ , K2′ , · · · , Km
) = (K1 − #(D1 ), K2 − #(D2 ), · · · , Km − #(Dm )).
For any y ∈ Vertex(X, σ), the grouping E1 (y), E2 (y), · · · , Em (y) of E of type
′
(y) of E ′ of type
(K1 , K2 , · · · , Km ) induces a grouping E1′ (y), E2′ (y), · · · , Em
′
′
′
′
(K1 , K2 , · · · , Km ), by setting Ej (y) = Ej (y)\Dj .
Apply Corollary 3.13 to the simplex (X, σ) and those groupings of E ′ , to obtain
′
(x) of E ′ for all newly
a subdivision (X, τ ) and groupings E1′ (x), E2′ (x), · · · , Em
introduced vertices x ∈ Vertex(X, τ ) such that the following hold.
For each newly introduced vertex x ∈ Vertex(X, τ ),
(2′ )
Ej′ (x) ⊂
[
Ej′ (y),
y∈Vertex(X,σ)
j = 1, 2, · · · , m.
For any simplex ∆ of (X, τ ) (after subdivision),
(3′ )
#(
\
x∈Vertex(∆)
E1′ (x)) ≥ K1′ −
n(n + 1)
,
2
where n = dim X.
Finally, let Ej (x) = Ej′ ∪ Dj for any x ∈ Vertex(X, τ ). Then the desired condition (1) of the corollary means Dj ⊂ Ej (x), which is true from the definition.
Also the conditions (2) and (3) of the corollary follows from (2′ ) and (3′ ).
⊔
⊓
Corollary 3.15. Suppose that (X, σ) is a simplicial complex. Suppose that
for each vertex x ∈ Vertex(X, σ), there is a grouping E1 (x), E2 (x), · · · , Em (x)
of E.
It follows that there is a subdivision (X, τ ) of (X, σ), and there is an extension
of the definition of the groupings of E for Vertex(X, σ) to the groupings of E
for Vertex(X, τ ) ⊃ Vertex(X, σ) such that the following properties hold.
For each newly introduced vertex x ∈ Vertex(X, τ ), if x ∈ ∆, where ∆ is a
simplex of (X, σ)(before subdivision), then
(1)
\
y∈Vertex(∆,σ)
Ej (y) ⊂ Ej (x),
j = 1, 2, · · · , m,
[
j = 1, 2, · · · , m.
and
(2)
Ej (x) ⊂
y∈Vertex(∆,σ)
Ej (y),
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
355
For any simplex ∆1 of (X, τ ) (after subdivision),
(3)
#(
\
x∈Vertex(∆1 )
E1 (x)) ≥ K1 −
n(n + 1)
,
2
where n = dim X.
(The above (1) and (2) imply that for any x ∈ Vertex(X, τ ),
\
[
Ej (y) ⊂ Ej (x) ⊂
Ej (y),
j = 1, 2, · · · , m.)
y∈Vertex(X,σ)
y∈Vertex(X,σ)
Proof: The proof is exactly the same as that of Corollaries 3.13 and 3.14. In
fact, in the proof of Corollary 3.13, we were working simplex by simplex from
the lowest dimension to the highest dimension. As same as Corollary 3.13,
when we work on simplex ∆, we should suppose that, we have already done
with ∂∆. The only difference is the following. We should choose the sets H i ,
Di differentlySaccording to the simplexTwe are working on. For simplex ∆,
choose Hi = y∈Vertex(∆) Ei (y), Di = y∈Vertex(∆) Ei (y), i = 1, 2 · · · , m.
⊔
⊓
Lemma 3.4 is a special case of the following theorem.
Theorem 3.16. Suppose that (X, σ) is a simplicial complex. Suppose that for
each vertex x ∈ Vertex(X, σ), there is a grouping E1 (x), E2 (x), · · · , Em (x) of
E.
It follows that there is a subdivision (X, τ ) of (X, σ), and there is an extension
of the definition of the groupings of E for Vertex(X, σ) to the groupings of E
for Vertex(X, τ ) ⊃ Vertex(X, σ) such that the following properties hold.
For each newly introduced vertex x ∈ Vertex(X, τ ), if x ∈ ∆, where ∆ is a
simplex of (X, σ) (before subdivision), then
\
(1)
Ej (y) ⊂ Ej (x),
j = 1, 2, · · · , m,
y∈Vertex(∆,σ)
and
(2)
Ej (x) ⊂
[
j = 1, 2, · · · , m.
Ej (y),
y∈Vertex(∆,σ)
For any simplex ∆1 of (X, τ ) (after subdivision),
(3)
#(
\
x∈Vertex(∆1 )
Ej (x)) ≥ Kj −
n(n + 1)
, j = 1, 2, · · · , m,
2
where n = dim X.
Proof: We will apply Corollary 3.15 to prove our theorem. First we can apply
Documenta Mathematica 7 (2002) 255–461
356
Guihua Gong
Corollary 3.15 to E1 and (X, σ) to make the condition (3) of the theorem hold
for E1 and any simplex of the subdivision, and also the conditions (1) and (2)
of the theorem hold. We call the simplicial structure after this step, τ1 .
Then we apply Corollary 3.15 to E2 (in place of E1 ) and (X, τ1 ) (in place
of (X, σ)). We call this new subdivision τ2 . Now (3) for E2 holds for any
simplex of new subdivision τ2 . Furthermore (1) and (2) of Corollary 3.15 hold
for (X, τ1 ) as the simplicial structure before subdivision (i.e., in place of (X, σ))
and (X, τ2 ) as the subdivision (i.e., in place of (X, τ )).
The important point is, (3) for E1 holds for any simplex ∆2 of (X, τ2 ), because
(3) for E1 holds for the simplex ∆1 of (X, τ1 ) which supports ∆2 (i.e., ∆1 ⊃ ∆2 ),
and because (1) holds for τ1 (in place of σ) and τ2 (in place of τ ). So, now (3)
holds for both E1 and E2 .
(It is also obvious that (1) for σ and τ2 ( in place of τ ) follows from (1) for σ
and τ1 (in place of τ ), together with (1) for τ1 (in place of σ) and τ2 (in place
of τ ). The same thing also holds for (2).)
Repeating this procedure, we can define τ3 , τ4 , and so on, until τm . Then (1),
(2), (3) hold for σ and τm and any Ej , j = 1, 2, · · · , m. Let τ = τm .
⊔
⊓
Remark 3.17. Let us remark that, in the proof of Lemma 3.11 when we
construct the sets E1 , simplex by simplex for (X, σ), it is impossible to obtain
\
dim(∆′ )(dim(∆′ ) + 1)
#
E1 (x) ≥ K1 −
,
2
x∈Vertex(∆′ )
for each simplex ∆′ of subdivision (X, τ ) of (X, σ). (Explained below.)
But from the proof of Corollary 3.12, we can make the following hold,
\
dim(∆)(dim(∆) + 1)
,
#
E1 (x) ≥ K1 −
2
′
x∈Vertex(∆ )
where ∆ is any simplex of (X, σ) which support ∆′ (i.e., ∆′ ⊂ ∆ as spaces).
In other words,
\
l(l + 1)
,
#
E1 (x) ≥ K1 −
2
′
x∈Vertex(∆ )
if ∆′ is a subspace of l-skeleton X (l) of (X, σ).
In the induction construction from dimension not larger than n−1 to dimension
n (see the proof of Lemma 3.11), in particular, from (∂∆0 × {t0 }, τ̃ ) to (∂∆0 ×
[t0 , t1 ], τ̃ ), for any simplex ∆ inside one of (∂∆0 × {t0 }, τ̃ ) and (∂∆0 × {t1 }, τ̃ ),
we do have
\
(n − 1)n
E1 (x) ≥ K1 −
,
2
x∈Vertex(∆)
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
357
from our construction (see condition (b) in the proof of Lemma 3.11). But for
simplices ∆ which are not completely sitting inside one of (∂∆0 × {t0 }, τ̃ ) and
(∂∆0 × {t1 }, τ̃ ), we do NOT have
\
x∈Vertex(∆)
E1 (x) ≥ K1 −
(n − 1)n
,
2
even if we assume dim(∆) ≤ n − 1.
For the application we have in mind, we need the following strengthened form
of Theorem 3.16 (in fact, we will need the version of the following result which
allow multiplicities; see Theorem 3.32).
Theorem 3.18. Let (X, σ) be a simplicial complex and Y = X (l) be the lskeleton of X. Suppose that there is a subdivision (Y, τ ) of (Y, σ) and a grouping
for each vertex of (Y, τ ) (and (X, σ)), such that
(a) if ∆ is a simplex of (Y, τ ), then
\
l(l + 1)
.
j = 1, 2, · · · , m;
#
Ej (y) ≥ Kj −
2
y∈Vertex(∆,τ )
(b) if ∆ is a simplex of (Y, σ) ⊂ (X, σ), and y ∈ ∆ is a vertex of (Y, τ ), then
\
[
Ej (x) ⊂ Ej (y) ⊂
Ej (x),
j = 1, 2, · · · , m.
x∈Vertex(∆,σ)
x∈Vertex(∆,σ)
It follows that there is a subdivision (X, τ̃ ) of (X, σ) and groupings for all the
vertices, such that
(1) (X, τ̃ )|Y = (Y, τ ), and groupings on Vertex(Y, τ ) are the same as the old
ones.
(2) if ∆ is a simplex of (X, σ), and x1 ∈ ∆ is a newly introduced vertex of
(X, τ̃ ), then
\
[
Ej (x) ⊂ Ej (x1 ) ⊂
Ej (x),
j = 1, 2, · · · , m;
x∈Vertex(∆,σ)
x∈Vertex(∆,σ)
′
(3) for each simplex ∆ of (X, τ̃ ), if ∆ is inside the l ′ -skeleton (X, σ)(l ) (l′ ≥ l)
of (X, σ), then
\
l′ (l′ + 1)
.
j = 1, 2, · · · , m.
#
Ej (x) ≥ Kj −
2
x∈Vertex(∆,τ̃ )
Proof: If one does not require (1) (i.e., if it is allowed to introduce more vertices
into (Y, τ )), then the theorem is Theorem 3.16 (see 3.17 also).
Documenta Mathematica 7 (2002) 255–461
358
Guihua Gong
Recall, in the proof of Theorem 3.16, we first constructed a subdivision (X, τ 1 )
and the groupings to make the above (3) hold for E1 . Then based on (X, τ1 ),
we constructed a new subdivision (X, τ2 ) and groupings to make the above (3)
hold also for E2 , and so on. If we use the same procedure to prove Theorem
3.18, we will encounter a difficulty in the second step. We have no problem
for the first step, since we can begin with what we already have on (X (l) , τ )
and work on each of the simplexes of dimension larger than l (see Lemma 3.11,
the proof of Corollary 3.12 and Remark 3.17). But for the second step, the
condition (a) may not hold for l-skeleton (X, τ1 )(l) of (X, τ1 ). So we need to
start with the simplex of the lowest dimension, which forced us to introduce
vertices on (Y, τ ) = (X (l) , τ ).
The following small trick can be used to avoid the difficulty mentioned above.
Consider simplex ∆. Suppose that the subdivision (∂∆, τ̃ ) and the groupings
for those vertices are chosen. Identify ∆ with ∂∆ × [0, 1]/∂∆ × {1} as in the
proof of Lemma 3.11. Choose a point t0 ∈ (0, 1), and write
∆ = ∂∆ × [0, t0 ] ∪ (∂∆ × [t0 , 1]/∂∆ × {1}).
Substitute ∆ by ∆sub = ∂∆ × [t0 , 1]/∂∆ × {1}. The simplicial structure τ̃pre
and the groupings on ∂∆sub = ∂∆ × {t0 } should be endowed the same as τ̃ and
the groupings on ∂∆ = ∂∆ × {0}. Then apply Theorem 3.16 to ∆sub . One may
introduce new vertices on (∂∆ × {t0 }, τ̃pre ), but no new vertices are introduced
on ∂∆ = ∂∆ × {0}. Finally, for the part ∂∆ × [0, t0 ], same as in the Step 4 of
the proof of Lemma 3.11, we apply Lemma 3.10 to make this part a simplicial
complex, in which we do not introduce any new vertices on ∂∆ × {0}.
⊔
⊓
3.19. For convenience, define
\
Ej (∆) =
Ej (x),
x∈Vertex(∆)
j = 1, 2, · · · , m
for each simplex ∆ of (X, τ̃ ). Then (3) of 3.18 becomes
#(Ej (∆)) ≥ Kj −
l′ (l′ + 1)
,
2
if ∆ is in the l′ -skeleton of (X, σ) (l′ ≥ l).
3.20. We need a different version of Theorem 3.18 which allows multiplicity.
Let w1 , w2 , · · · , wk be a k-tuple of positive integers. Let
E = {λ1∼w1 , λ2∼w2 , · · · , λk∼wk }
be an index set with multiplicity and λi 6= λj if i 6= j. (See 1.1.7 (b) for the
notation λ∼w .) Let w1 +w2 +· · ·+wk = K. (3.2 is a special case with each wi =
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
359
1.) Let K1 , K2 , · · · , Km be non negative integers with K1 +K2 +· · ·+Km = K.
Suppose that
∼pj1
Ej = {λ1
∼pj2
, λ2
∼pjk
, · · · , λk
},
j = 1, 2, · · · , m,
where pji are nonnegative integers. If {E1 , E2 , · · · , Em } satisfies
k
X
pji = Kj
for each j = 1, 2, · · · , m,
i=1
m
X
pji = wi
and
for each i = 1, 2, · · · , k,
j=1
then we call {E1 , E2 , · · · , Em } a grouping of E of type (K1 , K2 , · · · , Km ),
or just a grouping of E.
3.21. It is convenient to use the notations of union, intersection etc. for the
sets with multiplicity. A is called a subset of E if A is of the form
∼tk
∼t2
1
{λ∼t
1 , λ2 , · · · , λ k }
with 0 ≤ ti ≤ wi , for each 1 ≤ i ≤ k. Note that if all ti = 0, then A is called
the empty set. If ti = wi , then A = E. Let B be another subset of E of form
∼sk
∼s2
1
}. A is a subset of B (denoted by A ⊂ B) if ti ≤ si
{λ∼s
1 , λ2 , · · · , λ k
for all i. In general, define the union, intersection, and difference of
two subsets A and B of E as follows.
∼max(t1 ,s1 )
A ∪ B = {λ1
∼min(t1 ,s1 )
A ∩ B = {λ1
∼max(0,t1 −s1 )
A\B = {λ1
∼max(t2 ,s2 )
, λ2
∼max(tk ,sk )
, · · · , λk
, λ2
∼min(t2 ,s2 )
, · · · , λk
∼max(0,t2 −s2 )
, · · · , λk
, λ2
∼min(tk ,sk )
}
}
∼max(0,tk −sk )
}.
(The definitions of union and intersection can be easily generalized to finitely
many subsets of E.)
Warning 1: B ∩ (A\B) may be a nonempty set.
Warning 2: The assumption that {E1 , E2 , · · · Em } is a grouping of E does
′
NOT imply that ∪m
i=1 Ei = E or that Ei ∩ Ei′ = ∅ for i 6= i . (See 3.20.)
But A\B = A\(A ∩ B) still holds.
3.22. Let (X, σ) be a simplicial complex. Suppose that each vertex x ∈ X is
associated with a grouping {E1 , E2 , · · · , Em }, satisfying
#(Ej ) =
k
X
pji = Kj .
i=1
Documenta Mathematica 7 (2002) 255–461
360
Guihua Gong
(Recall that, for the notation of #, we count multiplicity.)
In Theorem 3.18, we introduced subdivisions (X, τ ) of (X, σ), and groupings
for newly introduced vertices to make #(Ej (∆)) large, for all simplices ∆ of
(X, τ ). In order to prove the decomposition theorem in the next section, we
need a stronger result, since the multiplicity of the spectrum of the homomorphism is involved. (See §2. We can not always perturb the map to have distinct
spectrum, like the one dimensional case.) Fortunately, this stronger result can
be proved in the same way as that for Theorem 3.18, with a few modifications.
For any subset F = {λ1∼u1 , λ2∼u2 , · · · , λk∼uk } ⊂ E, define
◦
1
2
k
F = {λ∼v
, λ∼v
, · · · , λ∼v
},
1
2
k
where
vi =
½
ui
0
if ui = wi
if ui < wi .
◦
That is, F is the set of all those elements λi , which are entirely inside F .
Evidently,
\
◦
◦
Ej (∆) =
Ej (x).
x∈Vertex(∆)
Instead of the condition that Ej (∆) is large (see 3.18 and 3.19), we need to
◦
◦
make Ej (∆) large for any simplex ∆ of (X, τ̃ ). For this purpose, Ej (x) should
be large for each vertex of (X, σ) at the beginning.
3.23. For each set F = {λ1∼u1 , λ2∼u2 , · · · , λk∼uk } ⊂ E, define
k
1
}
, λ2∼v2 , · · · , λ∼v
F̄ = {λ∼v
1
k
where
vi =
Obviously,
½
wi
0
if ui > 0
if ui = 0 .
◦
F ⊂ F ⊂ F̄ .
3.24. Let H1 , H2 , · · · , Hm (not necessarily disjoint) be finite subsets of E satisfying condition (∗) in 3.7, and E = H1 ∪ H2 ∪ · · · ∪ Hm . Suppose that
◦
Hi = Hi = H̄i
for each i = 1, 2, · · · , m.
In what follows, we will require that
Ej ⊂ H j ,
j = 1, 2, · · · , m
(comparing with the condition (2) in Theorem 3.18).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
361
3.25. For each subset I ⊂ {1, 2, · · · , m}, define
[
HI =
Hj .
j∈I
Let G′I = ∩j∈I Hj . Then define
GI = G′I \
[
J
⊃
6=
G′J .
I
Another way to define GI is by
GI = {λ ∈ E | λ ∈ Hi if and only if i ∈ I}.
(Note that GI may be an empty set for some I.) Obviously,
◦
◦
GI = GI = ḠI ⊂ HI = HI = H̄I .
If I ∩ J = ∅, then HI ∩ GJ = ∅.
Furthermore, for any λ ∈ E, there is a unique set I (defined by I = {i | λ ∈ Hi })
such that λ ∈ GI . Hence E is a disjoint union of
{GI , ∅ 6= I ⊂ {1, 2, · · · , m}}.
Similarly, we have
HI =
[
GJ .
J∩I6=∅
3.26. Under the above partition GI of E, two elements λ, µ ∈ E are in the
same part, if and only if the following is true. For any i = 1, 2, · · · , m, either
Hi contains both λ and µ, or Hi contains none of λ and µ.
For any E1 , E1′ ⊂ H1 , if #(E1 ∩ GI ) = #(E1′ ∩ GI ) for any I ⊂ {1, 2, · · · , m},
then from the end of 3.25,
X
X
#(E1 ∩ HI ) =
#(E1 ∩ GJ ) =
#(E1′ ∩ GJ ) = #(E1′ ∩ HI )
J∩I6=∅
J∩I6=∅
for any I ⊂ {1, 2, · · · , m}. Hence #(HI \E1 ) = #(HI \E1′ ). At this circumstance, either both of E1 and E1′ satisfy (∗∗) in 3.8, or both of them do not
satisfy (∗∗) in 3.8.
Note that Hi = ∪I∋i GI . A grouping {E1 , E2 , · · · Em } satisfies Ei ⊂ Hi (i =
1, 2, · · · m) if and only if for any i ∈
/ I, Ei ∩ GI = ∅.
For the rest of the section, let Ω ≥ max(w1 , w2 , · · · , wm ) be a fixed
number, where w1 , w2 , · · · , wm are the multiplicities in E. Note that for
our application, sometimes, we have to allow Ω to be larger than the maximum
multiplicity.
Documenta Mathematica 7 (2002) 255–461
362
Guihua Gong
Assumption 3.27. For each grouping {E1 (x), E2 (x), · · · , Em (x)}, we always
assume that
◦
#(Ej (x)) ≥ #(Ej (x)) − M Ω = Kj − M Ω,
j = 1, 2, · · · , m,
where M = 2m − 1. We not only require each initial grouping for (X, σ) to
satisfy the above assumption, but also require any new groupings for vertices
of (X, τ ) to satisfy the assumption.
Since M = 2m − 1, there are totally M non-empty subsets I ⊂ {1, 2, · · · , m}.
If the grouping {E1 , E2 , · · · , Em } satisfies
◦
#(Ej ∩ GI ) ≥ #(Ej ∩ GI ) − Ω
for all j = 1, 2, · · · , m and for all I ⊂ {1, 2, · · · , m}, then it also satisfies Assumption 3.27.
Lemma 3.28. If {E1 , E2 , · · · , Em } is a grouping of E with Ei ⊂ Hi , then there
′
is a grouping {E1′ , E2′ , · · · , Em
} of E satisfying Assumption 3.27, and
Ei′ ⊂ Hi ,
◦
Ei′ ⊃ Ei
for all i = 1, 2, · · · , m.
′
Proof: The proof is straight forward. Consider E1′ , E2′ , · · · , Em
to be m boxes
with no element at the beginning, and put each element of E into one of the
boxes, following the procedures described below.
◦
Step 1. Put all the elements of Ei into box Ei′ for each i = 1, 2, · · · , m. (Thus
◦
Ei′ ⊃ Ei .)
Step 2. Fix I ⊂ {1, 2, · · · , m}. For the set E1′ , if there is a λi ∈ GI \(E1′ ∪ E2′ ∪
′
· · · ∪ Em
) such that
#(E1′ ∩ GI ) + wi ≤ #(E1 ∩ GI ),
where wi is the multiplicity of λi in E, then put the entire set {λi∼wi } into E1′ .
(Note that if 1 ∈
/ I, then E1 ∩ GI = ∅. Hence for I, we need not do anything
for E1 .) Repeat this procedure until no such i exists. Thus, so far,
◦
#(E1′ ∩ GI ) = #(E1′ ∩ GI ) ≥ #(E1 ∩ GI ) − (Ω − 1).
For the same I above, repeat the above construction for the set E2′ , then E3′ ,
etc.
After this step has been completed for each I, (it is done for each set I separately) we have the following:
◦
#(Ej′ ∩ GI ) = #(Ej′ ∩ GI ) ≥ #(Ej ∩ GI ) − (Ω − 1).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
363
Step 3. Put what left for each GI from the previous steps, arbitrarily into the
boxes
′
E1′ , E2′ , · · · , Em
to make the following condition hold:
#(Ej′ ∩ GI ) = #(Ej ∩ GI ).
From the end of 3.26, Ei′ ⊂ Hi is a consequence of the above equation. (Note
′
that Ei ⊂ Hi .) Evidently, {E1′ , E2′ , · · · , Em
} is as desired.
⊔
⊓
3.29. A set E1 (⊂ H1 ) of K1 elements is said to satisfy the condition (∗ ∗ ∗), if
there is a grouping (E1 , E2 , · · · , Em ) of E (of type (K1 , K2 , · · · Km )), Ei ⊂ Hi ,
satisfying Assumption 3.27.
Obviously, (∗ ∗ ∗) implies (∗∗).
The following corollary is a direct consequence of Lemma 3.28.
Corollary 3.30. For any set E1 (⊂ H1 ) satisfying (∗∗), there is a set E1′ (⊂
H1 ) satisfying (∗ ∗ ∗) such that
◦
E1′ ⊃ E1 .
Proof: Since E1 satisfies (∗∗), we can extend E1 to a grouping {E1 , E2 , · · · , Em }
of E such that Ei ⊂ Hi for each i. By Lemma 3.28, there is a grouping
′
} satisfying Assumption 3.27, and Ei′ ⊂ Hi for each i. This is
{E1′ , E2′ , · · · , Em
condition (∗ ∗ ∗) for E1′ .
⊔
⊓
Lemma 3.31. Let E1 and F1 be two sets satisfying condition (∗ ∗ ∗). Suppose
that there is a λ ∈ E such that
◦
◦
{λ∼w } ⊂ F1 \E1 ,
where w is the multiplicity of λ in E. Then there are (perhaps repeating)
◦
elements µ1 , µ2 , · · · , µt ∈ E1 \F1 , where t = w − #({λ∼w } ∩ E1 ), such that
E1′ = (E1 ∪ {λ∼w })\{µ1 , µ2 , · · · , µt }
satisfies (∗∗) and
◦
◦
◦
#((E1′ ∩ E1 ) ∩ GI ) ≥ #(E1 ∩ GI ) − (w + Ω)
◦
≥ #(E1 ∩ GI ) − 2Ω
for each I ⊂ {1, 2, · · · , m}. As a consequence,
◦
◦
◦
#(E1′ ∩ E1 ) ≥ #(E1 ) − 2M Ω.
Documenta Mathematica 7 (2002) 255–461
364
Guihua Gong
Proof: Let t1 = #({λ∼w } ∩ E1 ). Then t1 < w. Applying Lemma 3.9 t :=
w − t1 times, one can obtain a (possibly repeating) set T ′ = {ν1 , ν2 , · · · , νt } ⊂
◦
E1 \F1 ⊂ E1 \F1 , such that
Ẽ1 = (E1 ∪ {λ∼w })\T ′
satisfies (∗∗).
◦
From 3.26, if another set T = {µ1 , µ2 , · · · , µt } ⊂ E1 \F1 , satisfies that
#(T ∩ GI ) = #(T ′ ∩ GI )
for each I ⊂ {1, 2, · · · , m}, then E1′ = (E1 ∪ {λ∼w })\T also satisfies (∗∗).
◦
T ⊂ E1 \F1 will be constructed to satisfy the following condition. For each
I ⊂ {1, 2, · · · , m},
#(T ∩ GI ) = #(T ′ ∩ GI ),
and (T ∩ (E1 \T )) ∩ GI is either empty or {µ∼s
i } for a certain µi ∈
{µ1 , µ2 , · · · , µt }. (Note that T ∩(E1 \T ) may not be empty, since we are dealing
with sets with multiplicities.)
To do the above, write
◦
∼s2
1
(E1 \F1 ) ∩ GI = {λ∼s
i1 , λi2 , · · · , }.
∼s2
1
Then put each of the sets {λ∼s
i1 }, {λi2 }, · · · , entirely into T one by one until
we can not do it without violating the restriction
#(T ∩ GI ) ≤ #(T ′ ∩ GI ).
∼s
Then make T to satisfy #(T ∩ GI ) = #(T ′ ∩ GI ) by putting part of {λij j }
into T if necessary.
Since #(T ) ≤ w, combining with the above condition for (T ∩ (E1 \T )) ∩ GI ,
after a moment thinking, one can obtain,
◦
#((E1 \T )◦ ∩ GI ) ≥ #(E1 ∩ GI ) − (w + Ω).
◦
◦
◦
(In fact, E1 \(E1 \T )◦ ⊂ T ∪ (T ∩ (E1 \T )), and (T ∪ (T ∩ (E1 \T ))) ∩ GI has at
most w + Ω elements.)
Hence
◦
◦
◦
#((E1′ ∩ E1 ) ∩ GI ) ≥ #(E1 ∩ GI ) − (w + Ω).
⊔
⊓
The following is the main result of this section. Together with Lemma 3.28, it
will be used in §4.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
365
Theorem 3.32. Let (X, σ) be a simplicial complex, and Y = X (l) , the lskeleton of X. Suppose that (Y, τ ) is a subdivision of (Y, σ) and, for each
vertex y ∈ Vertex(Y, τ ), there is a grouping E1 (y), E2 (y), · · · , Em (y) of E (of
type (K1 , K2 , · · · , Km )). Suppose that the groupings satisfy the following three
conditions:
(a) For each simplex ∆ of (Y, τ ), and i = 1, 2, · · · , m,
◦
#(Ei (∆)) ≥ Ki − (M Ω + M Ω dim Y · (dim Y + 1)),
where M = 2m − 1.
(b) Ei (x) ⊂ Hi , i = 1, 2, · · · , m, for each x ∈ Vertex(Y, τ ).
(c) Each grouping, for a vertex of (Y, τ ), satisfies Assumption 3.27.
It follows that there exist a subdivision (X, τ̃ ) of (X, σ) and a grouping for each
vertex of (X, τ̃ ), satisfying the following conditions.
(1) (X, τ̃ )|Y = (Y, τ ), and each grouping on Vertex(Y, τ ) is as same as the old
one.
(2) Ei (x) ⊂ Hi , i = 1, 2, · · · , m, for each x ∈ Vertex(X, τ̃ ), and if ∆ is a
simplex of (X, σ) (before the subdivision), and x ∈ ∆ is a newly introduced
vertex of (X, τ̃ ), then
\
◦
◦
Ej (y).
Ej (x1 ) ⊃
y∈Vertex(∆∩Y,τ )
′
(3) For each simplex ∆ of (X, τ̃ ), if ∆ is inside the l ′ -skeleton (X, σ)(l ) , (l′ >
l), of (X, σ), then
◦
#(Ej (∆)) ≥ Kj − (M Ω + M Ωl′ (l′ + 1)).
(4) Each grouping on Vertex(X, τ̃ ) satisfies Assumption 3.27.
Proof: (Sketch) The proof is the same as the one of 3.18 (see 3.11 to 3.18), using
Lemma 3.31 to replace Lemma 3.9. The arguments in 3.12 – 3.18 are easily
adopted in this new setting. We only give the proof for the part corresponding
to 3.11 and sketch the differences for other parts.
As in 3.11, consider only one simplex X = ∆0 with Y = ∂∆0 , and only one set
E1 (x).
Similar to Step 1 of 3.11, choose E1model to satisfy condition (∗ ∗ ∗) and
\
◦ model
◦
E1
⊃
E1 (x).
x∈Vertex(∂∆0 ,τ )
Replace (a) of 3.11 by
\
◦
◦ model
(E1 (x)∩E1
)
x∈Vertex(∂∆×{ti },τ̃ )
⊃
6=
\
x∈Vertex(∂∆×{ti−1 },τ̃ )
Documenta Mathematica 7 (2002) 255–461
◦
◦
(E1 (x)∩E1
model
) .
366
Guihua Gong
Keeping the notations in 3.11, in Step 2, replace G by
◦
◦
◦
◦
G = E1 (y1 , t0 ) ∩ E1 (y2 , t0 ) ∩ · · · ∩ E1 (yp , t0 ) ∩ E1
◦
If G = E1
model
model
.
, then define
E1 (yi , t1 ) = E1 model ,
◦
Suppose that G 6= E1
of λ. Then
model
i = 1, 2, · · · , p.
◦
. Choose λ ∈ E1
◦
{λ∼w } ⊂ E1
model
model
\G. Let w be the multiplicity
\G.
◦
For each (yi , t0 ), if λ ∈ E1 (yi , t0 ), then define E1 (yi , t1 ) = E1 (yi , t0 ) (as in Step
◦
2 of 3.11). If λ ∈
/ E1 (yi , t0 ), apply Lemma 3.31 to obtain E1′ satisfying(∗∗),
◦
E1′ ⊃ G ∪ {λ∼w } and
(A)
◦
◦
◦
#(E1′ ∩ E1 ) ≥ #(E1 ) − 2M Ω.
◦
◦
Then we can apply Corollary 3.29 to find E1′′ satisfying (∗ ∗ ∗) and E1′′ ⊃ E1′ .
Define
E1 (yi , t1 ) = E1′′ .
Then
◦
◦
E1 (yi , t1 ) ⊃ E1′ ⊃ G ∪ {λ∼w }.
The arguments in Step 3 and Step 4 of 3.11 can also be employed here. (Of
◦
course, at many places (not all places), one needs to replace Ei by Ei .) The
estimation (e) in Step 4 of 3.11 will be changed to
³◦
´
◦
◦
◦
◦
◦
# E1(u0 , t0 )∩ E1(u1 , t0 )∩· · ·∩ E1(ui , t0 )∩ E1(u0 , t1 )∩ E1(u1 , t1 )∩· · ·∩ E1(ui , t1 )
≥ K1 − [M Ω + M Ω dim Y · (dim Y + 1)] − 2M Ω(i + 1)
= K1 − [M Ω + M Ω · (n − 1) · n] − 2M Ω · n
= K1 − [M Ω + M Ω · n · (n + 1)].
(Here we used the above estimation (A) which is from Lemma 3.31.)
Since E1 (yi , t1 ) satisfies (∗ ∗ ∗), all the other parts (e.g., induction arguments)
in 3.11—3.18 can go through easily. In the part corresponding to the proof of
Corollary 3.14, the definition of Di should be changed to
Di =
\
◦
Ei (x).
x∈Vertex(∆)
⊔
⊓
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
367
We remark that in §4, we will only use the theorem of the case
that X = ∆, a single simplex with Y = ∂∆.
Remark 3.33. The condition Ei (x) ⊂ Hi in (2) can be strengthened as
[
Ei (x) ⊂
Ēi (y),
y∈Vertex(∆∩Y,τ )
where ∆ is a simplex of (X, σ) (before the subdivision) such that x ∈ ∆ is a
newly introduced vertex. (The notation is from 3.23.)
4
Decomposition Theorems
In this section, we will prove the decomposition theorems which are needed for
the proof of our main Reduction Theorem and the main results in [EGL]. The
following Theorem 4.1 is one version of the Decomposition Theorem. After
Theorem 4.1 has been proved, we will use [Li 2] to verify that the condition
of Theorem 4.1 holds for connecting homomorphisms φn,m (for each fixed n,
m should be large enough), with the maps a1 , a2 , · · · aL (see below) factoring
through interval [0, 1] or the single point space {pt}. In such a way, we can
prove our main decomposition theorems (Theorem 4.35 and Theorem 4.37).
Theorem 4.1. Let X be a connected finite simplicial complex, and F ⊂ C(X)
be a finite set which generates C(X). For any ε > 0, there is an η > 0 such
that the following statement is true.
Suppose that a unital homomorphism φ : C(X) → P MK ′ (C(Y ))P (rank(P ) =
K) (where Y is a finite simplicial complex) satisfies the following condition:
There are L continuous maps
a1 , a2 , · · · , aL : Y −→ X
such that for each y ∈ Y, SPφy and Θ(y) can be paired within η, where
Θ(y) = {a1 (y)∼T1 , a2 (y)∼T2 , · · · , aL (y)∼TL }
and T1 , T2 , · · · , TL are positive integers with
T1 + T2 + · · · + TL = K = rank(P ).
(See 1.1.7(b) for notation x∼Ti .) Let T = 2L (dim X + dim Y )3 . It follows that
there are L mutually orthogonal projections p1 , p2 , · · · , pL ∈ P MK ′ (C(Y ))P
such that
PL
(i) kφ(f )(y) − p0 (y)φ(f )(y)p0 (y) ⊕ i=1 f (ai (y))pi (y)k < ε, for any f ∈ F and
PL
y ∈ Y , where p0 = P − i=1 pi ;
(ii) kp0 (y)φ(f )(y) − φ(f )(y)p0 (y)k < ε for any f ∈ F and y ∈ Y ;
(iii) rank(pi ) ≥ Ti − T for 1 ≤ i ≤ L, and hence rank(p0 ) ≤ LT .
Documenta Mathematica 7 (2002) 255–461
368
Guihua Gong
4.2. In the above theorem, some of the pi may be zero projections if Ti ≤ T .
But when the theorem is applied later in this article, the positive integers T i
are always very large compared with T = 2L (dim X + dim Y )3 .
The proof of this theorem will be divided into several small steps. The results
in §2 and §3 will be used. In fact, §2 and §3 will only be used in the proof of
Theorem 4.1, no other place in this paper or [EGL]. (The results in §2 have
some other applications.)
4.3. The theorem is trivial if X = {pt}, applying Theorem 1.2 of [Hu,Chapter
8]. Without loss of generality, we assume that X 6= {pt}. By the results in
§2, we can assume that φ has maximum spectral multiplicity at most Ω :=
dim X + dim Y .
4.4.
For any ε > 0, there is an η > 0 such that for any x1 , x2 ∈ X, if
dist(x1 , x2 ) < 2η, then
|f (x1 ) − f (x2 )| <
ε
3
for all f ∈ F.
We will prove that this η is as desired.
4.5.
Recall, from 1.2.5, for any positive integer n, P n X is the symmetric
product of n-copies of X. Also, any element Λ ∈ P K X can be considered as a
set with multiplicity. So SPφy ∈ P K X.
Suppose that Λ1 ∈ P k1 X, Λ2 ∈ P k2 X, · · · , Λt ∈ P kt X. Write
Λ1 = {λ1 , λ2 , · · · , λk1 }
Λ2 = {λk1 +1 , λk1 +2 , · · · , λk1 +k2 }
..
.
Λt = {λk1 +···+kt−1 +1 , · · · , λk1 +···+kt }
as sets with multiplicity. By abusing the notation, we use {Λ1 , Λ2 , · · · , Λt } to
denote
{λ1 , λ2 , · · · , λk1 , λk1 +1 , · · · , λk1 +k2 , · · · · · · , λk1 +k2 +···+kt } ,
which defines an element in P k1 +k2 +···+kt X.
(Note that {Λ1 , Λ2 , · · · , Λt } = Λ1 ∪ Λ2 ∪ · · · ∪ Λt , if {Λj } are mutually disjoint.
See 3.21 for the definition of unions of sets with multiplicity.)
4.6. For any fixed point y ∈ Y , write
SPφy = {λ1∼w1 , λ2∼w2 , · · · , λk∼wk }
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
369
with λi 6= λj if i 6= j. Note that w1 + w2 + · · · + wk = K. We denote the above
k by ty to indicate that this integer depends on y. Define
(a)
θ(y) =
1
min dist(λi , λj ).
4 1≤i<j≤ty
Then θ(y) > 0 for any y ∈ Y . (Of course, θ(y), in general, does not depend on
y continuously.)
For each i = 1, 2, · · · , ty , there is an open set U (y, i) ∋ λi such that
µ
¶
η
(b)
diameter(U (y, i)) ≤ min
, θ(y) .
2(dim Y + 1)
Then, obviously,
(c)
dist(U (y, i), U (y, j)) ≥ 2θ(y)
if
i 6= j.
Applying Lemma 1.2.10, there is an (connected) open neighborhood O(y) of y
such that
[
[
[
SPφy′ ⊂ U (y, 1) U (y, 2) · · · U (y, ty )
for all y ′ ∈ O(y). Define the continuous maps
Λ1 : O(y) −→ P w1 U (y, 1) (⊂ P w1 X)
Λ2 : O(y) −→ P w2 U (y, 2) (⊂ P w2 X)
..
.
Λty : O(y) −→ P wty U (y, 1) (⊂ P wty X)
by Λi (y ′ ) = SPφy′ ∩ U (y, i). Then
SPφy′ = {Λ1 (y ′ ), Λ2 (y ′ ), · · · , Λty (y ′ )}
for each y ′ ∈ O(y). Later on, we will use the disjoint open cover
[
[
[
U (y, 1) U (y, 2) · · · U (y, ty ) ⊃ SPφy′
of SPφy′ to decompose SPφy′ into a disjoint union of SPφy′ ∩ U (y, t) and
to identify the elements in each set SPφy′ ∩ U (y, t) as a single element with
multiplicity wt .
We further require that O(y) is so small that
(c)
diameter(ai (O(y))) ≤
η
,
2(dim Y + 1)
where ai : Y → X is any one of the continuous maps a1 , a2 , · · · , aL , appeared
in Theorem 4.1.
Documenta Mathematica 7 (2002) 255–461
370
Guihua Gong
4.7. Considering the open cover {O(y)}y∈Y of Y , where the open sets O(y)
are from 4.6, there exists a finite sub-cover
O = {O1 , O2 , · · · , O• } ⊂ {O(y)}y∈Y ,
of Y .
Without loss of generality, we assume that the simplicial complex structure
(Y, σ) of Y satisfies the following condition, because we can always refine it if
necessary.
For each simplex ∆ of (Y, σ), the closure of
Star(∆) :=
[
interior(∆′ )
∆′ ∩∆6=∅
can be covered by an open set Oi ∈ O, where
interior(∆′ ) = ∆′ \∂∆′ .
(Note that Star(∆) is an open set, see 1.4.2)
4.8. For each y ∈ Y , in order to construct p1 (y), p2 (y), · · · , pL (y), as in Theorem 4.1, we need to split SPφy into L sets E1 (y), E2 (y), · · · , EL (y) such that
each set Ei (y) is contained in an open ball of ai (y) with small radius (smaller
than 2η) and that #(Ei (y)) = Ti , 1 ≤ i ≤ L, where ai and Ti are maps
and positive integers appeared in Theorem 4.1. Since E1 (y), E2 (y), · · · , EL (y)
may have non-empty intersection (because of the multiplicity of the spectrum),
◦
◦
we need to introduce certain subsets of them, which are E1 ⊂ E1 (y), E2 ⊂
◦
E2 (y), · · · , EL ⊂ EL (y), in the notations of 3.22. This will become precise when the index set with multiplicity is introduced later. The projections
p1 (y), p2 (y), · · · , pL (y), to be constructed, will be certain sub-projections of the
◦
◦
◦
spectral projections corresponding to E1 , E2 , · · · , EL , respectively. (See Definition 1.2.8 for the spectral projection.)
Following §3, a split of SPφy into L sets E1 (y), E2 (y), · · · , EL (y) will be called
a grouping of SPφy . The word “grouping” is reserved only for this
purpose.
Recall, from 4.6, SPφy′ can be written as a disjoint union
SPφy′ = (SPφy′ ∩ U (y, 1))
[
(SPφy′ ∩ U (y, 2))
[
···
[
(SPφy′ ∩ U (y, ty )).
And the elements in each set SPφy′ ∩U (y, t) can be identified as a single element
with multiplicity. This will serve as the index set for the groupings. To avoid
confusion, the above decomposition is NOT called a “grouping” of SPφ y′ . It
is called a decomposition instead.
In the next few paragraphs, we apply §3 to construct a subdivision (Y, τ ) of
(Y, σ) and useful groupings for all vertices y ∈ Vertex(Y, τ ).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
371
4.9. Let ∆ be a simplex of (Y, σ) and
O(y1 ), O(y2 ), · · · , O(yi ),
the list of all open sets in O, each of which covers ∆. Suppose that
θ(y1 ) ≤ θ(y2 ) ≤ · · · ≤ θ(yi ).
From 4.6, for any y ∈ ∆ ⊂ ∩ik=1 O(yk ), and j ∈ {1, 2, · · · , i},
[
[
[
SPφy ⊂ U (yj , 1) U (yj , 2) · · · U (yj , tyj ).
Claim: If j < j ′ ∈ {1, 2, · · · , i}, then each open set U (yj , t) (t = 1, 2 · · · , tyj )
ty
′
j
intersects with at most one of {U (yj ′ , s)}s=1
.
Proof of the Claim: Suppose that the claim is not true, that is, for some
t ∈ {1, 2, · · · , tyj }, there are two different s1 , s2 ∈ {1, 2 · · · , tyj ′ } such that
U (yj , t) ∩ U (yj ′ , s1 ) 6= ∅
and
U (yj , t) ∩ U (yj ′ , s2 ) 6= ∅.
Together with the fact that diameter(U (yj , t)) ≤ θ(yj ) (see (b) in 4.6), it yields
dist(U (yj ′ , s1 ), U (yj ′ , s2 )) ≤ θ(yj ).
This contradicts with (c) in 4.6 which gives
dist(U (yj ′ , s1 ), U (yj ′ , s2 )) ≥ 2θ(yj ′ ) > θ(yj ).
(Recall θ(yj ) ≤ θ(yj ′ ).) This proves the claim.
Still suppose that j < j ′ . From the claim, we have the following. For each
y ∈ ∆, if two different elements of SPφy are identified as a single element in
the decomposition
[
[
[
U (yj , 1) U (yj , 2) · · · U (yj , tyj )
(i.e., if these two elements are in the same open set U (yj , t) for some t ∈
{1, 2, · · · , tyj }), then these two elements are also identified as a single element
in the decomposition
[
[
[
U (yj ′ , 1) U (yj ′ , 2) · · · U (yj ′ , tyj ′ )
(i.e., these two elements are also in the same open set U (yj ′ , s) for some s ∈
{1, 2, · · · , tyj ′ }).
Therefore, the decompositions of SPφy corresponding to y1 and yi are the finest
and coarsest decompositions, respectively, among all the above decompositions
(corresponding to y1 , y2 , · · · , yi ). The coarsest decomposition will be used to
decompose SPφy into several sets. The elements in each of the sets will be
Documenta Mathematica 7 (2002) 255–461
372
Guihua Gong
identified as a single element with multiplicity. Denote θ(yi ) by θ(∆). (Recall
that O(y1 ), O(y2 ), · · · , O(yi ) is the list of all open sets in O, each of which
covers ∆. Therefore, θ(yi ) — the maximum of all {θ(yj )}ij=1 — depends only
on ∆.)
Introduce the following notations.
Λ(∆, 1)(y) = U (yi , 1) ∩ SPφy ,
Λ(∆, 2)(y) = U (yi , 2) ∩ SPφy ,
..
.
Λ(∆, t∆ )(y) = U (yi , tyi ) ∩ SPφy ,
where t∆ = tyi . Recall (see 4.6) that SPφyi is written as
SPφyi = {λ1∼w1 , λ2∼w2 , · · · , λk∼wk },
where k = t∆ = tyi . Since y ∈ ∆ ⊂ O(yi ),
1 ≤ t ≤ t∆ ,
#(Λ(∆, t)(y)) = wt ,
counting multiplicity. Define set
Λ(∆) = {Λ(∆, 1)∼w1 , Λ(∆, 2)∼w2 , · · · , Λ(∆, k)∼wk },
where k = t∆ . That is, identify all the elements of SPφy in Λ(∆, t)(y) as a
single element (denoted by Λ(∆, t)) with the multiplicity.
As above, we will use Λ(∆, t)(y) for two purposes. It is a subset of SPφy , or it
is a single element in Λ(∆) which repeats wt times.
Strictly speaking, wt (t = 1, 2, · · · , t∆ ) should be written as wt (∆), and the set
Λ(∆) should be written as
{Λ(∆, 1)∼w1 (∆) , Λ(∆, 2)∼w2 (∆) , · · · , Λ(∆, k)∼wk (∆) }. When there is a danger of
confusion, we will use wt (∆) instead of wt .
4.10.
Let Y ′ ⊂ Y be a path connected subspace. Usually we will let Y ′
be either an open or a closed subset. Suppose that there are positive integers
u1 , u2 , · · · , ut and continuous maps
A(Y ′ , i) : Y ′ → P ui X,
i = 1, 2, · · · , t,
such that {SPφy }y∈Y ′ can be decomposed as
SPφy = {A(Y ′ , 1)(y), A(Y ′ , 2)(y), · · · , A(Y ′ , t)(y)}
for all y ∈ Y ′ . We say that the above decomposition of {SPφy }y∈Y ′
satisfies the condition (S) (S stands for separation) if
(S): there are mutually disjoint open sets U1 , U2 , · · · , Ut ⊂ X satisfying
A(Y ′ , i)(y) ⊂ Ui ,
∀y ∈ Y ′ , i = 1, 2, · · · , t.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
Define
373
A(Y ′ ) = {A(Y ′ , 1)∼u1 , A(Y ′ , 2)∼u2 , · · · , A(Y ′ , t)∼ut },
where ui = #(A(Y ′ , i)), counting multiplicity. Again, A(Y ′ , s)(y) is used for
two purposes. It is regarded as a subset of SPφy or as a single element of A(Y ′ )
with multiplicity us .
In fact, if U1 , U2 , · · · Ut are open sets, with mutually disjoint closure, such that
SPφy ⊂ U1 ∪ U2 ∪ · · · ∪ Ut , ∀y ∈ Y ′ , then #(SPφy ∩ Ui ), y ∈ Y ′ are constants,
denoted by ui , (note that Y ′ is path connected). Furthermore, the maps
A(Y ′ , i) : Y ′ → P ui X, i = 1, 2, · · · t,
defined by A(Y ′ , i)(y) = SPφy ∩ Ui , are continuous, and they determine a
decomposition of {SPφy }y∈Y ′ as
SPφy = {A(Y ′ , 1)(y), A(Y ′ , 2)(y), · · · , A(Y ′ , t)(y)}
satisfying the condition (S). (See Lemmas 1.2.9 and 1.2.10.)
For any y ∈ Y ′ , a grouping E1 , E2 , · · · , EL of SPφy induces a UNIQUE grouping
A(Y ′ )
E1
A(Y ′ )
, E2
A(Y ′ )
Ei
A(Y ′ )
, · · · EL
of A(Y ′ ), defined by
= {A(Y ′ , 1)∼v1 , A(Y ′ , 2)∼v2 , · · · , A(Y ′ , t)∼vt }, i = 1, 2, · · · , L,
where vj = #(Ei ∩ A(Y ′ , j)(y)), counting multiplicity. (Here the intersection
of sets is defined as for the sets with multiplicity, as in 3.21.)
On the other hand, let E1 , E2 , · · · , EL be a grouping of A(Y ′ ). Define a grouping of the set SPφy , for any y ∈ Y ′ , in the following way. For any j = 1, 2, · · · , L,
if the part Ej (for the grouping of A(Y ′ )) contains exactly w elements of
{A(Y ′ , s)∼us } (w ≤ us ), then the part Ej (for the grouping of the set of
SPφy ) contains exactly w elements (counting multiplicity) which are contained
in A(Y ′ , s)(y). Since these w elements are to be chosen, the induced grouping
is not unique. But we will always fix one of them for use.
Let E1 , E2 , · · · , EL be a grouping of
A(Y ′ ) = {A(Y ′ , 1)∼u1 , A(Y ′ , 2)∼u2 , · · · , A(Y ′ , t)∼ut }.
◦
◦
◦
Define E1 , E2 , · · · , EL as in 3.22.
Although the subsets of SPφy corresponding to Ei are not unique, the subsets
◦
◦
of SPφy corresponding to Ei are unique. We denote them by Ei |y . Also,
◦
◦
◦
◦
#(Ei ) = #(Ei |y ) counting multiplicity. Note that we use Ei |y instead of Ei (y)
for the following reason (also see the next paragraph). We reserve the notation
{Ei (y)}Li=1 for the grouping of SPφy which is associated to a vertex y in a
certain simplicial complex (Y, τ ). (τ is a subdivision of σ.)
Suppose that y ∈ Y ′ . Let E1 (y), E2 (y), · · · EL (y) be a grouping of SPφy . Then
A(Y ′ )
A(Y ′ )
A(Y ′ )
it induces a grouping E1
(y), E2
(y), · · · EL
(y) of A(Y ′ ) as above.
◦
◦
A(Y ′ )
A(Y ′ )
The sets Ei
(y) are well defined as subsets of A(Y ′ ). (Warning: Ei
(y)
Documenta Mathematica 7 (2002) 255–461
374
Guihua Gong
◦
A(Y )
are not subsets of SPφy .) Also, from the last paragraph, the sets Ei
(y)|z
are well defined as subsets of SPφz for any z ∈ Y ′ (may be different from y).
◦
◦
A(Y ′ )
A(Y ′ )
Furthermore, #(Ei
(y)) = #(Ei
(y)|z ).
In the next few paragraphs, each grouping of SPφy can be referred as a grouping
of A(Y ′ ) for different space Y ′ and different decomposition A(Y ′ ) provided that
y ∈ Y ′ , or vise versa.
∆
(in 4.9) could be regarded
4.11. Note that the collection of sets {Λ(∆, i)}ti=1
as a decomposition of SPφy , y ∈ ∆ (see 4.6 also). And this decomposition
satisfies condition (S) for ∆ in place of Y ′ ; therefore 4.10 can be applied to
Λ(∆) as A(Y ′ ).
As mentioned in 4.8, we will introduce the groupings of SPφy for all vertices
of a certain subdivision of (Y, σ). As in section 3, for a simplex ∆ of (Y, σ),
once we have the subdivision (∂∆, τ ) of (∂∆, σ) and groupings for all vertices in
Vertex(∂∆, τ ), then we can define the subdivision (∆, τ ) of (∆, σ) and introduce
the groupings for all newly introduced vertices. One may notice that, in section
3, for different vertices, the index sets involved are the same. But in the setting
here, the index sets SPφy are different for different vertices y. So some
special care should be taken.
Suppose that (∆, σ) is a simplicial complex consisting of a single simplex ∆ and all its faces. Suppose that there is a subdivision (∂∆, τ ) of
(∂∆, σ) and the groupings E1 (y), E2 (y), · · · , EL (y) of SPφy for all vertices
y ∈ Vertex(∂∆, τ ) (see notation in 3.1–3.3). Attention: When we introduce a grouping E1 (z), E2 (z), · · · , EL (z) of SPφz for any newly introduced vertex z ∈ interior(∆) = ∆\∂∆, the following procedure will always be
used.
First, as in 4.10, we can regard the groupings E1 (y), E2 (y), · · · , EL (y) of
Λ(∆)
Λ(∆)
Λ(∆)
SPφy as groupings E1
(y), E2
(y), · · · , EL (y) of Λ(∆) for all vertices
Λ(∆)
Λ(∆)
(y ′ )∩· · ·, as a subset of Λ(∆),
(y)∩Ei
y ∈ Vertex(∂∆, τ ). (Then the set Ei
◦
Λ(∆)
makes sense, for vertices y, y ′ , · · · ∈ Vertex(∂∆, τ ). Also {Ei
(y)}Li=1 are subsets of Λ(∆).) Then we use these groupings of the same index set, Λ(∆), applying the results from section 3 (see 3.32), to introduce subdivision (∆, τ ) of (∆, σ)
and groupings of Λ(∆) for all newly introduced vertices z ∈ ∆\∂∆. Finally,
theses groupings of Λ(∆) will induce the groupings E1 (z), E2 (z), · · · , EL (z) of
SPφz as in 4.10 (not unique, but we fix one of them for our use). Furthermore,
◦
◦
◦
Λ(∆)
Λ(∆)
Λ(∆)
as in 4.10, E1
(z), E2
(z), · · · , EL (z) are well defined subsets of Λ(∆)
◦
◦
◦
Λ(∆)
Λ(∆)
Λ(∆)
and E1
(z)|z′ , E2
(z)|z′ , · · · , EL (z)|z′ are well defined subsets of SPφz′
for any z ′ ∈ ∆ (not necessarily a vertex).
4.12. Let ∆′ be a face of ∆. Then for y ∈ ∆′ ⊂ ∆, both Λ(∆′ ) and Λ(∆) can
be viewed as decompositions of SPφy . Recall, in 4.9, the decomposition corresponding to Λ(∆) is the coarsest decomposition among those corresponding to
O(yj ) such that O(yj ) ⊃ ∆ and that O(yj ) ∈ O. Since ∆′ ⊂ ∆, any open set
in O which covers ∆ will also cover ∆′ . Therefore, the decomposition of SPφy
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
375
corresponding to Λ(∆′ ) is coarser than that corresponding to Λ(∆). That is,
each set Λ(∆′ , s)(y) is a finite union of certain sets,
[
[
Λ(∆, t1 )(y) Λ(∆, t2 )(y)
··· ,
as subsets of SPφy . (Notice that if ws′ = ws (∆′ ) is the multiplicity appeared
in Λ(∆′ ) for Λ(∆′ , s) and wt = wt (∆) is the multiplicity appeared in Λ(∆) for
Λ(∆, t), then ws′ = wt1 + wt2 + · · · , a finite sum.)
It follows that if y ∈ ∆′ ⊂ ∆ and E1 (y), E2 (y), · · · , EL (y) is a grouping of
SPφy , then
◦
Λ(∆′ )
Ej
◦
Λ(∆)
(y)|y′ ⊂ Ej
(y)|y′ ,
regarded as subsets of SPφy′ for any y ′ ∈ ∆′ . Now we are ready to construct
the subdivision (Y, τ ) of (Y, σ), and the grouping for each vertex of (Y, σ) and
each vertex of the complex (Y, τ ) (after subdivision).
Since the notations y1 , y2 , · · · , y• have been already used for the open cover
O = {O(y1 ), O(y1 ), · · · , O(y• )}, we use z1 , z2 , · · · to denote the points in Y ,
especially the vertices of certain simplicial structure.
4.13.
Let Ω = dim X + dim Y, M = 2L − 1, where L is the number of
the continuous maps {ai } appeared in the statement of 4.1. Note that all the
multiplicities wt appearing in any of Λ(∆) do not exceed Ω, by 4.3 and the
construction of Λ(∆) (see 4.6 and 4.9).
For each vertex z ∈ Vertex(Y, σ), by the condition of Theorem 4.1, SPφz and
Θ(z) = {a1 (z)∼T1 , a2 (z)∼T2 , · · · , aL (z)∼TL }
can be paired within η. Therefore, we can define a grouping
Epre,1 (z), Epre,2 (z), · · · , Epre,L (z) of SPφz , with T1 , T2 , · · · , TL elements, respectively, counting multiplicity, such that
(1)
dist(λ, ai (z)) < η
if λ ∈ Epre,i (z), where η is as in 4.4. (We denote them by Epre,i because this
grouping will be modified later.)
We can regard such a grouping of SPφz as a grouping of Λ(∆), where ∆ ∋ z is
a simplex.
First we regard it as the grouping of Λ({z}), where {z} is the 0-dimensional
simplex of (Y, σ) corresponding to vertex z. By Lemma 3.28, we can modify
the grouping to satisfy the Assumption 3.27. Then this modified grouping of
Λ({z}) could induce a grouping on SPφz , for which the condition (1) above
may not hold. But if we carefully choose the sets Hi in Lemma 3.28, we could
still guarantee that any elements λ ∈ Ei are close to ai (z) (see (2) below). In
this subsection, we will also introduce the sets Hi (∆) to serve as the sets Hi of
Lemma 3.28 and Theorem 3.32, when we construct groupings on ∆ from the
groupings on ∂∆, by applying Theorem 3.32.
Documenta Mathematica 7 (2002) 255–461
376
Guihua Gong
For each vertex z0 ∈ Vertex(Y, σ), the notation {z0 } is used to denote the
corresponding zero dimensional simplex of (Y, σ). The above grouping induces
Λ({z })
a grouping {Epre,i0 } of
∼wt{z
Λ({z0 }) = {Λ({z0 }, 1)∼w1 , Λ({z0 }, 2)∼w2 , · · · , Λ({z0 }, t{z0 } )
0}
}.
Define subsets H1 ({z0 }), H2 ({z0 }), · · · , HL ({z0 }) of Λ({z0 }) as follows. For any
i = 1, 2, · · · , L, Hi ({z0 }) is the collection of all
Λ({z0 }, t)∼wt (⊂ Λ({z0 })) satisfying
¾
½
1
Λ({z0 }, t)(z0 ) ⊂ x : dist(x, ai (z0 )) < η +
·η .
(dim Y + 1)
Note that each set Λ({z0 }, t)(z0 ) (as a subset of certain U (yj , t), from 4.9) has
diameter at most 2(dimηY +1) < (dimηY +1) (see (b) in 4.6). Combining this fact
with (1) above, we know that if λ1 ∈ Epre,i (z0 ) and λ1 ∈ Λ({z0 }, t)(z0 ), then
(2)
dist(λ, ai (z0 )) < η +
η
(dim Y + 1)
Λ({z })
for any λ ∈ Λ({z0 }, t)(z0 ). That means Epre,i0 ⊂ Hi ({z0 }).
By Lemma 3.28, the above grouping can be modified to another grouping
Λ({z })
Ei 0 of Λ({z0 }) satisfying
◦
#(Ei
Λ({z0 })
) ≥ Ti − M Ω.
Λ({z })
(This is Assumption 3.27.) And Ei 0 ⊂ Hi ({z0 }) still holds, regarded as a
grouping of Λ({z0 }).
The
above
grouping
of
Λ({z0 })
could
induce
a
grouping
E1 (z0 ), E2 (z0 ), · · · , EL (z0 ) of SPφz0 (see 4.10 and 4.11). This grouping
will be used as the grouping for vertex z0 . Even though (1) may not hold for
λ in the new Ei (z0 ), (2) holds for any λ in the new Ei (z0 ), from the definition
Λ({z })
of Hi ({z0 }), and Ei 0 ⊂ Hi ({z0 }).
For each simplex ∆ of (Y, σ), let us also define the subsets H1 (∆), H2 (∆), · · · ,
HL (∆) of Λ(∆) as follows. For each j = 1, 2, · · · , L, Hj (∆) is the collection of
all such Λ(∆, t)∼wt (⊂ Λ(∆)) that Λ(∆, t)(z), as a subset of SPφz , satisfies
Λ(∆, t)(z) ⊂ {x : dist(x, ai (z)) < η +
dim(∆) + 1
· η}
(dim Y + 1)
for any z ∈ ∆. These sets will serve as the sets H1 , H2 , · · · , HL when we apply
Theorem 3.32.
The following fact follows directly from the definition of Λ(∆, t) and Hi (∆),
which will be used in 4.14:
Suppose that z ∈ ∆. A grouping E1 , E2 , · · · EL of SPφz , regarded as a grouping
of Λ(∆), satisfies Ei ⊂ Hi (∆) if and only if for any λ ∈ Ei (as a subset of
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
377
SPφz ), for the index t satisfying λ ∈ Λ(∆, t)(z) (such t exists; see 4.9), we have
{Λ(∆, t)∼wt (∆) } ⊂ Hi (∆).
4.14. Beginning with the simplicial structure (Y, σ) and the above groupings
for all 0-dimensional simplex (i.e., vertex) of (Y, σ), we will construct a subdivision (Y, τ ) of (Y, σ) and the groupings for newly introduced vertices. We
will refine (Y, σ), simplex by simplex, from the lowest dimension to the highest
dimension by use of Theorem 3.32.
To avoid confusion, use Γ, Γ1 , Γ2 , Γ′ , etc. to denote the simplices of (Y, τ ), after subdivision, and reserve the notations ∆, ∆′ , ∆1 , etc. for the simplices of
(Y, σ)—with original simplicial complex structure σ introduced in 4.7.
As the induction assumption, we suppose that there are a subdivision (∂∆, τ )
of (∂∆, σ) and the groupings of SPφz for all vertices z ∈ Vertex(∂∆, τ ) with
the following properties.
(1) If ∆′ is a proper face of (∆, σ) (by a proper face of ∆, we mean a face
∆′ with ∆′ ⊂ ∂∆) and z ∈ ∆′ , then the grouping of Λ(∆′ ), induced by the
grouping of SPφz satisfies
Ei ⊂ Hi (∆′ ).
Λ(∆′ )
In other words, Ei
(z) ⊂ Hi (∆′ ).
(2) Let Γ be a simplex of (∂∆, τ ) with vertices z0 , z1 , · · · , zj . If Γ ⊂ ∆′ , where
∆′ is a proper face of ∆, then
³◦
´
◦
◦
Λ(∆′ )
Λ(∆′ )
Λ(∆′ )
# Ei
(zj )
(z1 ) ∩ · · · ∩ Ei
(z0 ) ∩ Ei
≥ Ti − [M Ω + M Ω dim ∆′ (dim ∆′ + 1)]
(≥ Ti − [M Ω + M Ω dim ∂∆(dim ∂∆ + 1)]) .
(3) For each vertex z of (∂∆, τ ), Assumption 3.27 holds. I.e.,
◦
Λ(∆′ )
#(Ei
(z)) ≥ Ti − M Ω
for any proper face ∆′ of ∆ with z ∈ ∆′ .
Λ(∆′ )
(In the above conditions (1), (2) and (3), {Ei
(z)}Li=1 are regarded as group′
ings of the set Λ(∆ ) (with multiplicity); see 4.11.)
Now we define the subdivision (∆, τ ) of (∆, σ) and the groupings for all newly
introduced vertices. The restriction of the simplicial structure (∆, τ ) on ∂∆
will be the same as (∂∆, τ ), that is, we will only introduce new vertices inside
interior(∆) = ∆\∂∆. We need to define the groupings as groupings of Λ(∆).
Then they will induce groupings of SPφz .
Claim: For any vertex z of (∂∆, τ ), if the grouping of SPφz is regarded as the
grouping of Λ(∆), then
Λ(∆)
(1’) Ei (z) ⊂ Hi (∆) for i = 1, 2, · · · , L. In other words, Ei
(z) ⊂ Hi (∆).
Proof of the Claim: Let y be the point yi in the definition of Λ(∆) in 4.9. Then
∆ ⊂ O(y) ∈ O. (We avoid the notation yi , since i is used for Ei above. So we
use y instead.)
Documenta Mathematica 7 (2002) 255–461
378
Guihua Gong
Let λ ∈ Ei (z). Since z ∈ ∂∆, there is a proper face ∆′ of ∆ such that
z ∈ ∆′ . By (1) above, Ei (z) ⊂ Hi (∆′ ) (regarded as a grouping of Λ(∆′ )).
From the end of 4.13, there is an index s such that λ ∈ Λ(∆′ , s)(z) and that
′
{Λ(∆′ , s)∼ws (∆ ) } ⊂ Hi (∆′ ).
Recall, from 4.12, Λ(∆′ , s)(z ′ ) is a finite union Λ(∆, t1 )(z ′ ) ∪ Λ(∆, t2 )(z ′ ) ∪ · · ·,
for any z ′ ∈ ∆′ ⊂ ∆. Hence, there is an index t such that
λ ∈ Λ(∆, t)(z) ⊂ Λ(∆′ , s)(z),
where both Λ(∆, t)(z) and Λ(∆′ , s)(z) are regarded as subsets of SPφz . To
prove the claim, by the end of 4.13, we only need to prove {Λ(∆, t)∼wt (∆) } ⊂
Hi (∆). From the definition of Hi (∆), this is equivalent to
(A)
Λ(∆, t)(z1 ) ⊂ {x : dist(x, ai (z1 )) < η +
dim(∆) + 1
· η}
(dim Y + 1)
′
for any z1 ∈ ∆. From {Λ(∆′ , s)∼ws (∆ ) } ⊂ Hi (∆′ ) and the definition of Hi (∆′ ),
we have
Λ(∆′ , s)(z ′ ) ⊂ {x : dist(x, ai (z ′ )) < η +
dim(∆′ ) + 1
· η}
(dim Y + 1)
for any z ′ ∈ ∆′ . In the above, if we choose z ′ = z—the vertex in the claim—
(and note that Λ(∆, t)(z) ⊂ Λ(∆′ , s)(z)), then
(a)
Λ(∆, t)(z) ⊂ {x : dist(x, ai (z)) < η +
dim(∆′ ) + 1
· η}.
(dim Y + 1)
On the other hand, from (d) in 4.6, we have
(b)
diameter(ai (∆)) ≤ diameter(ai (O(y))) <
η
.
2(dim Y + 1)
And from (b) in 4.6, we have
(c)
diameter(U (y, t)) <
η
.
2(dim Y + 1)
From 4.6 and 4.9, Λ(∆, t)(z1 ) ⊂ U (y, t) for any z1 ∈ ∆ ⊂ O(y). Combining
this with (c) above, for any µ ∈ Λ(∆, t)(z1 ) (z1 ∈ ∆), we have
dist(µ, Λ(∆, t)(z)) <
η
.
2(dim Y + 1)
Then combining it with (a) above, we have
dist(µ, ai (z)) < η +
η
(dim(∆′ ) + 1)η
+
.
(dim Y + 1)
2(dim Y + 1)
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
379
Finally, combining it with (b),
dist(µ, ai (z1 )) < η +
(dim(∆′ ) + 1)η
η
η
+
+
(dim Y + 1)
2(dim Y + 1) 2(dim Y + 1)
≤η+
dim(∆) + 1
,
(dim Y + 1)
since dim(∆′ ) ≤ dim(∆) − 1. Note that z1 ∈ ∆ and µ ∈ Λ(∆, t)(z1 ) are
arbitrary, this proves (A) and the Claim.
Suppose that Γ is a simplex of (∂∆, τ ) with vertices z0 , z1 , · · · zj . Suppose that
Γ ⊂ ∆′ , where ∆′ is a face of ∆. As mentioned in 4.12,
◦
Λ(∆′ )
Ei
◦
Λ(∆)
(z)|z′ ⊂ Ei
(z)|z′
as a subset of SPφz′ for all z ′ ∈ ∆′ and all z = z0 , z1 , · · · , zj . Therefore, from
(2) and³ (3) above, we have the following (2’) and
´ (3’).
◦
Λ(∆)
(2’) # Ei
◦
Λ(∆)
(z0 ) ∩ Ei
◦
Λ(∆)
(z1 ) ∩ · · · ∩ Ei
(zj )
³◦
´
◦
◦
Λ(∆)
Λ(∆)
Λ(∆)
= # Ei
(z0 )|z′ ∩ Ei
(z1 )|z′ ∩ · · · ∩ Ei
(zj )|z′
³◦
´
◦
◦
Λ(∆′ )
Λ(∆′ )
Λ(∆′ )
(z0 )|z′ ∩ Ei
(z1 )|z′ ∩ · · · ∩ Ei
(zj )|z′
≥ # Ei
³◦
´
◦
◦
Λ(∆′ )
Λ(∆′ )
Λ(∆′ )
= # Ei
(z0 ) ∩ Ei
(z1 ) ∩ · · · ∩ Ei
(zj )
≥ Ti − [M Ω + M Ω dim ∂∆(dim ∂∆ + 1)],
for every simplex Γ ⊂ (∂∆, τ ) with vertices z0 , z1 , · · · , zj .
(3’) The Assumption 3.27 holds for the grouping {Ei (z)}L
i=1 regarded as a
grouping of Λ(∆), i.e.,
◦
Λ(∆)
#(Ei
(z)) ≥ Ti − M Ω,
where z is a vertex of (∂∆, τ ).
Apply Theorem 3.32 to obtain a subdivision (∆, τ ) of (∆, σ), and, for each
newly introduced vertex z ∈ ∆, a grouping E1 (z), E2 (z), · · · EL (z) of SPφz
such that (1), (2) and (3) hold with the version obtained by replacing ∆′ by ∆,
and dim(∂∆) by dim ∆. (As mentioned in 4.11, for each vertex z, we should
first get the groupings of Λ(∆), then this grouping induces a grouping of SPφ z .)
Using Mathematical Induction, combined with 4.13, we obtain our subdivision
(Y, τ ) of (Y, σ) and the groupings.
We summarize what we obtained in 4.13 and 4.14 as in the following proposition.
Proposition: There is a subdivision (Y, τ ) of (Y, σ), and for all vertices
z ∈ Vertex(Y, τ ), there are groupings E1 (z), E2 (z), · · · , EL (z) of SPφz of type
Documenta Mathematica 7 (2002) 255–461
380
Guihua Gong
(T1 , T2 , · · · , TL ) (i.e., #(Ei (z)) = Ti ∀i) such that the following are true.
(1) If ∆ is a simplex of (Y, σ) (before subdivision) and z ∈ ∆, then the
Λ(∆)
Λ(∆)
Λ(∆)
grouping (E1
(z), E2
(z), · · · EL (z)) of Λ(∆), induced by the grouping
(E1 (z), E2 (z), · · · , EL (z)) of SPφz , satisfies
Λ(∆)
Ei
(z) ⊂ Hi (∆).
(2) Let Γ be a simplex of (Y, τ ) with vertices z0 , z1 , · · · , zj . If Γ ⊂ ∆, where ∆
is a simplex of (Y, σ) (before subdivision), then
´
³◦
◦
◦
Λ(∆)
Λ(∆)
Λ(∆)
(z0 ) ∩ Ei
(z1 ) ∩ · · · ∩ Ei
(zj )
# Ei
≥ Ti − [M Ω + M Ω dim ∆(dim ∆ + 1)]
(≥ Ti − [M Ω + M Ω dim Y (dim Y + 1)]) .
(We do not need the condition (3) any more.)
4.15. For the simplicial complex (Y, τ ), there is a finite open cover
{W (Γ) : Γ is a simplex of (Y, τ )}
of Y , with the following properties.
(a) W (Γ) ⊃ interior(Γ) = Γ\∂Γ.
(b) If W (Γ1 ) ∩ W (Γ2 ) 6= ∅, then either Γ1 is a face of Γ2 or Γ2 is a face of Γ1 .
(Such open cover has been constructed in 1.4.2 (b).)
For any simplex Γ, we will construct an open set O(Γ) ⊃ Γ and introduce a
decomposition Ξ(Γ) of {SPφy }y∈O(Γ) , which is the finest possible decomposition
satisfying the condition (S) for Γ in place Y ′ in 4.10.
Recall that K = rank(P ), and y 7→ SPφy defines a map SPφ : Y → P K X. We
will prove the following easy fact.
Claim 1: SPφ|Γ := ∪z∈Γ SPφz (⊂ X) has at most K connected components.
(For K = 1, the claim says that the image of a connected space Γ under a
continuous map SPφ : Γ → P 1 X = X is connected. This is a trivial fact.)
Proof of Claim 1: Suppose that by the contrary, SPφ|Γ has more than
K connected components. Write SPφ|Γ = X1 ∪ X2 ∪ · · · ∪ XK+1 , where
X1 , X2 , · · · , XK+1 are mutually disjoint non empty closed subsets (which are
not necessary connected).
There are open sets U1 , U2 , · · · , UK+1 with mutually disjoint closures such that
Ui ⊃ Xi . Then for any z ∈ Γ, SPφz ⊂ ∪K+1
i=1 Ui . By Lemma 1.2.9, for each i,
#(SPφz ∩ Ui ) is a nonzero constant. Hence #(SPφz ) ≥ K + 1, contradicting
with #(SPφz ) = K = rank(P ), counting multiplicity. This proves the claim.
We are back to our construction of open set O(Γ) and decomposition Ξ(Γ).
Write SPφ|Γ = X1 ∪ X2 ∪ · · · ∪ Xt , where X1 , X2 , · · · , Xt (with t ≤ K) are mutually disjoint connected components of SPφ|Γ . Choose open sets U1 , U2 , · · · , Ut
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
381
with mutually disjoint closures such that Xi ⊂ Ui . By Lemma 1.2.9, there is
an open set O(Γ) ⊃ Γ such that SPφ|O(Γ) ⊂ ∪ti=1 Ui .
As in 4.10, define
Ξ(Γ, t)(z) = SPφz ∩ Ui , ∀z ∈ O(Γ), i = 1, 2, · · · , t.
This gives a decomposition
SPφz = {Ξ(Γ, 1)(z), Ξ(Γ, 2)(z), · · · , Ξ(Γ, t)(z)}, ∀z ∈ O(Γ).
Let ci = #(Ξ(Γ, i), counting multiplicity. And write
Ξ(Γ) := {Ξ(Γ, 1)∼c1 , Ξ(Γ, 2)∼c2 , · · · , Ξ(Γ, t)∼ct }.
Note that the above decomposition satisfies condition (S) in 4.10 as the decomposition of spectrum on O(Γ) (not only on Γ). In 4.16 below, when we apply
4.10, we will use U (Γ) (a subset of O(Γ)) in place of Y ′ of 4.10. Obviously,
Ξ(Γ) is the finest decomposition among all the decompositions of (SPφ z )z∈Γ
satisfying condition (S) on Γ, since each Xi is connected. In particular, if
z ∈ Γ ⊂ ∆, where ∆ is a simplex of (Y, σ) (before subdivision), then the decomposition of SPφz corresponding to Ξ(Γ) is finer than the decomposition of
SPφz corresponding to Λ(∆).
We will use the following fact later.
Claim 2: If Γ′ ⊂ Γ is a face, then for any z ∈ O(Γ′ ) ∩ O(Γ), the decomposition of SPφz corresponding to Ξ(Γ′ ) is finer than the decomposition of SPφz
corresponding to Ξ(Γ).
Proof of Claim 2: The Claim follows from the definition of Ξ(Γ) and the fact
that any connected component of SPφ|Γ′ is completely contained in a connected
component of SPφ|Γ .
4.16. For each simplex Γ, define U (Γ) = W (Γ) ∩ O(Γ).
{U (Γ); Γ is a simplex of (Y, τ )} is an open covering of Y since U (Γ) ⊃
interior(Γ).
For each U = U (Γ), we will define mutually orthogonal projection valued functions
P1U , P2U , · · · , PLU : U (Γ) ∋ y 7→ { sub-projections of P (y)}.
Then apply Proposition 3.2 of [DNNP] to construct the globally defined projections p1 , p2 , · · · , pL for our Theorem 4.1.
(Attention: For each vertex z of (Y, τ ), we have a grouping
E1 (z), E2 (z), · · · , EL (z) of SPφz . It will induce a grouping of Ξ(Γ), as in 4.10,
if Γ ∋ z. In the following construction of PiU (y), this grouping will be used.
That is, we will use the decomposition of SPφz corresponding to Ξ(Γ). The decomposition of SPφz corresponding to Λ(∆) will NOT be used in the definition
of PiU (y) at all— it is only used in the estimation of rank(PiU ).
In the definition of the grouping E1 (z), E2 (z), · · · , EL (z) of SPφz , it involves the
decomposition of SPφz corresponding to Λ(∆). But once it has been defined, it
Documenta Mathematica 7 (2002) 255–461
382
Guihua Gong
makes sense by itself without the decomposition of SPφz corresponding to Λ(∆)
◦
Λ(∆)
as a reference (though Ei
(z)|z only makes sense with the decomposition as
the reference).)
Back to our construction. For each y ∈ U (= U (Γ)) and each i = 1, 2, · · · , L,
define PiU (y) to be the spectral projection of φy corresponding to
³◦
´
◦
◦
Ξ(Γ)
Ξ(Γ)
Ξ(Γ)
Ei (z0 ) ∩ Ei (z1 ) ∩ · · · ∩ Ei (zj ) |y ,
◦
Ξ(Γ)
where z0 , z1 , · · · , zj are all vertices of Γ, and the notations Ei (z) and
◦
◦
◦
Ξ(Γ)
Ξ(Γ)
Ξ(Γ)
Ei (z)|y are as in 4.10. (That is, Ei (z) is a subset of Ξ(Γ) and Ei (z)|y
is a subset of SPφy .)
By Lemma 1.2.9, the above functions PiU (y) depend on y continuously. In
fact, for each i and any y ∈ U (Γ) ⊂ Y , PiU (y) is the spectral projection of
φy corresponding to an open subset (of X)— in the notation of 4.15 (see the
paragraph after the proof of Claim 1 in 4.15), the open subset is the union of
all open subsets Uj ⊂ X such that
◦
Ξ(Γ)
Ξ(Γ, j) ∈ Ei
◦
Ξ(Γ)
(z0 ) ∩ Ei
◦
Ξ(Γ)
(z1 ) ∩ · · · ∩ Ei
(zj )(⊂ Ξ(Γ)).
(Note that when we apply Lemma S
1.2.9, we use the fact that {Ui } have mutually disjoint closures and SPφy ⊂ Ui from 4.15.) Recall, the decomposition
of SPφz corresponding to Ξ(Γ) is finer than any decomposition of SPφz corre◦
◦
Λ(∆)
Ξ(Γ)
(z0 )|z , regarded as
sponding to Λ(∆), if Γ ⊂ ∆. Therefore, Ei (z0 )|z ⊃ Ei
a subset of SPφz , for any vertex z0 ∈ Γ and any point z ∈ Γ. By Condition (2)
of the grouping (see 4.14),
rank(PjU ) ≥ Tj − [M Ω + M Ω dim Y (dim Y + 1)]
for each U .
The projections PiU , i = 1, 2, · · · , L are mutually orthogonal, since they are
spectral projections corresponding to mutually disjoint subsets of X.
Let Γ′ be a face of Γ and z ∈ U (Γ) ∩ U (Γ′ ). By Claim 2 in 4.15, opposite to
the case of decompositions corresponding to Λ(∆) and Λ(∆′ ), the decomposition of SPφz corresponding to Ξ(Γ′ ) is finer than that corresponding to Ξ(Γ).
Therefore,
◦
◦
Ξ(Γ)
Ξ(Γ′ )
Ei (z0 )|y ⊂ Ei
(z0 )|y
for all z0 ∈ Vertex(Γ′ , τ ) ⊂ Vertex(Γ, τ ). Combining it with the fact that
Vertex(Γ′ , τ ) ⊂ Vertex(Γ, τ ), we get
\
\
′
◦
◦
Ξ(Γ )
Ξ(Γ)
Ei
(zj ) |y .
Ei (zj ) |y ⊂
zj ∈Vertex(Γ′ ,τ )
zj ∈Vertex(Γ,τ )
Consequently,
U (Γ)
Pi
U (Γ′ )
(y) ≤ Pi
(y)
if y ∈ U (Γ) ∩ U (Γ′ ).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
383
Finally, from the condition (1) of the groupings (see the proposition in the end
of 4.14) and the definition of Hi (∆), we have
³◦
o
´
n
Ξ(Γ)
(dim Y +1)
Ei (z) |y ⊂ λ; dist(λ, ai (y)) < η + (dim
Y +1) · η = 2η ,
where ∆ is any simplex of (Y, σ) satisfying Γ ⊂ ∆. Therefore, PiU (y) is the
spectral projection of φy corresponding to a subset of
{λ; dist(λ, ai (y)) ≤ 2η} ⊂ X.
We have proved the following lemma.
Lemma 4.17. There is a collection U of finitely many open sets which covers
Y . For each open set U ∈ U, there are mutually orthogonal projection valued
continuous functions
P1U , P2U , · · · , PLU : U ∋ y 7→ { sub-projections of P (y)}
with the following properties.
(1) If U1 , U2 ∈ U, and U1 ∩ U2 6= ∅, then either
PiU1 (z) ≤ PiU2 (z)
is true for all i = 1, 2, · · · , L and all z ∈ U1 ∩ U2 , or
PiU2 (z) ≤ PiU1 (z)
is true for all i = 1, 2, · · · , L and all z ∈ U1 ∩ U2 .
(2) rank(PiU (z)) ≥ Ti − [M Ω + M Ω dim Y (dim Y + 1)].
(3) Each PiU (z) is a spectral projection of φz corresponding to a subset of
{λ; dist(λ, ai (z)) < 2η} .
4.18. For i = 1, 2, · · · , L, applying Proposition 3.2 of [DNNP] to {PiU }U ∈U ,
there exist continuous projection valued functions
U
U
pU
1 , p2 , · · · , pL : Y ∋ y 7→ { sub-projections of P (y)}
such that
pi (y) ≤
and that
_
{PiU (y); y ∈ U ∈ U}
rank(pi ) ≥ Ti − [M Ω + M Ω dim Y (dim Y + 1)] − dim Y > Ti − T .
(Note that T = 2L (dim X + dim Y )3 .)
Documenta Mathematica 7 (2002) 255–461
384
Guihua Gong
By Condition (1) of 4.17, for each y,
span{PiU ; y ∈ U ∈ U} = PiU0
for a certain U0 ∋ y which does not depend on i. Therefore, {pi (y)}Li=1 are
mutually orthogonal since {PiU0 }Li=1 are mutually orthogonal.
PL
4.19. We will prove that the above projections {pi }Li=1 and p0 = P − i=1 pi
are as desired in Theorem 4.1. This is a routine calculation, as in the proof of
Theorem 2.7 of [GL1] or the last part of the proof of Theorem 2.21 of [EG2].
(See 1.5.4 and 1.5.7 also.) Since we need an extra property of p0 φp0 (described
in 4.20 below), we write down the complete proof.
For each y ∈ Y , as mentioned in 4.18, there exists an open set U0 ∈ U with
U0 ∋ y such that
span{PiU ; y ∈ U ∈ U} = PiU0 ,
i = 1, 2, · · · , L.
Let Pi (y) = PiU0 (y). Then
pi (y) ≤ Pi (y),
i = 1, 2, · · · , L,
and each Pi (y) is the spectral projection corresponding to a certain subset of
{λ; λ ∈ SPφy , dist(λ, ai (y)) < 2η} .
Let mutually different elements µ1 , µ2 , · · · , µs ∈ SPφy be the list of spectra which are not in the set of those spectra belonging to the projections {Pi (y)}Li=1 . Let q1 , q2 , · · · , qs be spectral projections corresponding to
{µ1 }, {µ2 }, · · · , {µs }, respectively. (The rank of each qi is the multiplicity of µi
in SPφy .) Then
s
L
X
X
qi .
Pi (y) +
P (y) =
i=1
i=1
Therefore,
p0 (y) = P (y) −
L
X
pi (y) =
L
X
i=1
i=1
s
X
(Pi (y) − pi (y)) +
qi .
i=1
Since the spectra belonging to Pi (y) are within distance 2η of ai (y), by the way
η is chosen in 4.4, for each f ∈ F ,
kφ(f )(y) − [
L
X
f (ai (y))Pi +
i=1
s
X
ε
.
3
f (µi )qi ]k <
i=1
Therefore, for each f ∈ F , kp0 (y)φ(f )(y) − φ(f )(y)p0 (y)k <
(∗)
kp0 (y)φ(f )(y)p0 (y) − [
L
X
i=1
f (ai (y))(Pi (y) − pi (y)) +
Documenta Mathematica 7 (2002) 255–461
2ε
3 ,
s
X
i=1
and
f (µi )qi ]k <
ε
.
3
Simple Inductive Limit C ∗ -Algebras, I
385
Also, for all f ∈ F ,
kpi (y)φ(f )(y) − f (ai (y))pi (y)k <
ε
3
kφ(f )(y)pi (y) − f (ai (y))pi (y)k <
Let P ′ =
PL
i=1
and
ε
.
3
pi . Then
PL
kP ′ (y)φ(f
PL )(y)p0 (y)k = k i=1 pi (y)φ(f )(y)p0 (y)k PL
≤ k i=1 [pi (y)φ(f )(y) − pi (y)f (ai (y))]p0 (y)k + k i=1 pi (y)f (ai (y))p0 (y)k
≤ 3ε + 0 = 3ε .
Similarly, for all f ∈ F ,
kp0 (y)φ(f )(y)P ′ (y)k <
ε
.
3
Also,
kP ′ (y)φ(f )(y)P ′ (y) −
L
M
f (ai (y))pi (y)k <
i=1
ε
.
3
Combining all the above estimations, we have, for f ∈ F ,
kφ(f )(y) − p0 (y)φ(f )(y)p0 (y) ⊕
L
M
f (ai (y))pi (y)k <
i=1
ε ε ε
+ + = ε.
3 3 3
This ends the proof of Theorem 4.1.
⊔
⊓
Attention: In fact, we proved that the conclusion of Theorem 4.1 holds not
only for f in the finite set F , but also for any f satisfying the condition that if
dist(x, x′ ) < 2η, then kf (x) − f (x′ )k < 3ε .
Remark 4.20. The following is the (∗) from 4.19:
(∗)
kp0 (y)φ(f )(y)p0 (y) − [
L
X
i=1
f (ai (y))(Pi (y) − pi (y)) +
s
X
i=1
f (µi )qi ]k <
ε
.
3
Recall that for any x, x′ ∈ X, if dist(x, x′ ) < 2η, then
kf (x) − f (x′ )k <
ε
3
for all f ∈ F .
|
Note that
by
PξLy : C(X) → p0 (y)M• (C)p0 (y),
Pdefined
s
ξy (f ) = i=1 f (ai (y))(Pi (y) − pi (y)) + i=1 f (µi )qi , is a homomorphism. By
1.2.18, we have the following claim.
Documenta Mathematica 7 (2002) 255–461
386
Guihua Gong
Claim: Let {x1 , x2 , · · · , xr } be an η-dense subset of X. Suppose that mutually
orthogonal projections p1 , p2 , · · · , pr ∈ (P − p0 )MK ′ (C(Y ))(P − p0 ) satisfy
rank(pi ) ≥ rank(p0 ).
Let ψ : C(X) → (p0 ⊕ p1 ⊕ p2 ⊕ · · · ⊕ pr )MK ′ (C(Y ))(p0 ⊕ p1 ⊕ p2 ⊕ · · · ⊕ pr )
be the positive linear map defined by
ψ(g) = p0 φ(g)p0 ⊕
r
X
g(xi )pi ,
i=1
for all g ∈ C(X). Then ψ(F ) is weakly approximately constant to within ε.
This fact will be used later.
Remark 4.21.
The proof of Theorem 4.1 is very long and complicated.
We point out that the following direct approaches will encounter difficulties.
(These discussions have appeared in §1.5.)
1. One may let PiU (y) be the spectral projections corresponding to the open
sets
{λ; dist(λ, ai (y)) < η}
and make use of Proposition 3.2 of [DNNP] to construct the projection pi .
The trouble is that such {pi }Li=1 are not mutually orthogonal since PiU are not
mutually orthogonal.
2. For each sufficiently small neighborhood U , applying the theorem about
spectral multiplicity from §2, one can construct mutually orthogonal projections {PiU (y)}Li=1 with relatively large rank such that each PiU (y) is the spectral
projection corresponding to a subset of
{λ; dist(λ, ai (y)) < η}.
W
But one cannot guarantee that the projection
to {PiU ; U ∋ y} is
W associated
orthogonal to the projection associated to {PjU ; U ∋ y}, for i 6= j. So one
still can not obtain orthogonal projections {pi }Li=1 .
3. One may try to define p1 , p2 , · · · , pL , one by one. For example, after p1 (y) is
defined, try to choose P2U (y) to be orthogonal to p1 (y) and to be the spectral
projection of a certain subset of X. Then this subset can not be chosen to be
a subset of {λ; dist(λ, a2 (y)) < η} since some spectra may have been taken
out when p1 (y) is defined. In fact, this subset can be chosen to be a subset
of {λ; dist(λ, a2 (y)) < 2η}. In this way, when we define PiU (y), it will be a
spectral projection corresponding to a subset of
{λ; dist(λ, ai (y)) < i · η}.
In order for the theorem to hold, L·η needs to be small, which makes η depend
on L. This is not useful at all for the application.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
387
Remark 4.22. Note that in 4.19, when we prove that the projections {pi }L
i=1
satisfy the desired conditions (i) and (ii) of Theorem 4.1, we only use the
property that for any y ∈ Y , pi (y), i = 1, 2, · · · , L, are subprojections of
Pi (y), i = 1, 2, · · · , L, respectively. This means, (i) and (ii) of Theorem 4.1 hold
′
for any set of projections {p′i }L
i=1 with pi ≤ pi , i = 1, 2, · · · , L. So we have the
freedom to replace any pi by its subprojection (with suitable rank). This fact
is important for the discussion below and in 4.41 and 4.44.
In what follows, we will use the fact that, for the projections in MK ′ (C(Y )) of
rank at least dim(Y ), cancellation always holds. That is , if three projections
p, q and r in M• (C(Y )) satisfy that rank(p) > dim(Y ), rank(q) > dim(Y ) and
p ⊕ r is Murry von Neumann equivalent to q ⊕ r, then p is Murry von Neumann
equivalent to q.
(a) In fact, in 4.19, rank(pi ) for our projections pi satisfy the stronger condition
(see 4.18):
rank(pi ) ≥ Ti − [M Ω + M Ω dim Y (dim Y + 1)] − dim Y.
From Theorem 1.2 of [Hu, Chapter 8], there is a trivial projection p′i < pi such
that
rank(p′i ) ≥ rank(pi ) − dim Y
≥ Ti − [M Ω + M Ω dim Y (dim Y + 1)] − 2 dim Y.
That is, rank(p′i ) is still larger than Ti − T , where T = 2L (dim X + dim Y )3 .
(In fact it is larger than Ti − T + 2 dim Y .) In Theorem 4.1, replacing pi by p′i ,
one makes all the projections {pi }Li=1 trivial.
(b) Suppose that there is an i0 ∈ {1, 2, · · · , L} such that Ti0 > T + dim Y .
Suppose that the projections p1 , p2 , · · · , pL are trivial as in (a). In particular,
suppose that rank(pi0 ) ≥ Ti0 − T + 2 dim Y as mentioned in (a). By [Hu],
P ∈ MK ′ (C(Y )) (the total projection of the target algebra P MK ′ (C(Y ))P in
Theorem 4.1) can be written in the form
q ⊕ (trivial projection),
where q is of rank Ti0 − T + dim Y . It follows from [Hu], that there is a subprojection p′i0 of pi0 which is unitarily equivalent to q. Replacing pi0 by p′i0 ,
and keeping all the other projections pi , then P will be unitarily equivalent to
a projection of the form
L
M
i=1
pi ⊕ (trivial projection).
LL
Therefore, p0 = P − i=1 pi is a trivial projection. (Note that rank(p0 ) ≥
dim Y .)
In other words, in Theorem 4.1, we can choose all the projections p0 , p1 , · · · , pL
to be trivial except one of them, pi0 , where i0 6= 0. In particular, p0 is a trivial
projection, as comparing with (a) above.
Documenta Mathematica 7 (2002) 255–461
388
Guihua Gong
The following theorem is proved in [EGL].
Theorem 4.23. ([EGL]) Let A = lim (An , φn,m ) be an inductive limit C ∗ n→∞
algebra (not necessarily unital) with
An =
tn
M
M[n,i] (C(Xn,i )),
i=1
where Xn,i are simplicial complexs. Then one can write A = lim (Bn , ψn,m )
n→∞
with
tn
M
M{n,i} (C(Yn,i )),
Bn =
i=1
where Yn,i are (not necessarily connected) simplicial complexs, with dim(Yn,i ) ≤
dim(Xn,i ), such that all the connecting maps ψn,m are injective.
Furthermore, if (An , φn,m ) satisfies the very slow dimension growth condition,
then so does (Bn , ψn,m ).
4.24. Without loss of generality, in the rest of this article, we
will assume that the connecting maps φn,m in the inductive limit
system are injective. Without this assumption one can still prove all the
theorems in this paper by modifying our arguments, and by passing to some
good subsets of Xn,i . But this assumption makes the discussions much simpler.
As mentioned inL1.1.5, we will suppose that the inductive limit algebra
tn
A = lim (An = i=1
M[n,i] (C(Xn,i )), φn,m ) satisfies the very slow dimension
n→∞
growth condition.
4.25.
As a consequence of Theorem 4.1 and the lemma inside 1.5.11— a
result due to Li—, one can obtain a decomposition for each (partial map of a)
connecting map φi,j
n,m (m large enough), with the major part factoring though
an interval algebra. But for our application, we need a certain part of the
decomposition to be defined by point evaluations and (even if it is not large
absolutely) to be relatively large compared to the “bad” part p0 φp0 , where p0
is the projection in Theorem 4.1, and φ is the map corresponding to φi,j
n,m (see
i
.
1.2.18 and 1.2.19), i.e., φ = φi,j
|
n,m e11 An e11
Following Section 2 of [Li3] (see the proof of Theorem 2.28 in [Li3]), we can
prove our main Decomposition Theorem (see Theorem 4.37 below). [Li3] only
proves the special case that Xn,i = graphs (one dimensional spaces). Although
the idea behind Li’s proof is reasonably simple and clear (see the explanation in
2.29 of [Li3]), the proof itself is complicated and long. It combines several difficulties together. For convenience in the higher dimensional case, we will give a
slightly different approach. (See 1.5.25 for the explanation of the difference between our approach and Li’s approach.) Our proof will be a little shorter, and
perhaps easier to follow (hopefully). More importantly, using this approach, we
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
389
will be able to prove the Decomposition Theorem for any homomorphism provided that the homomorphism satisfies a certain quantitative condition (see
Theorem 4.35 below). (Li’s theorem is for the homomorphism φn,m with m
sufficiently large.) This slightly stronger version of the theorem is needed in
[EGL] to prove the Uniqueness Theorem. It should be emphasized that our
proof is essentially the same as Li’s proof in spirit.
The idea behind our proof is roughly as follows.
In [Li2, 2.18–2.19] (see 1.5.11), Li proves that for fixed η > 0, for m large
enough, and for any (partial) connecting map φi,j
n,m —denoted by φ—, there
are L continuous maps β1 , β2 , · · · , βL : Y (= Xm,j ) −→ X (= Xn,i ), factoring
through the interval [0, 1], such that for each y ∈ Y , the set SPφy and the set
©
ª
Θ(y) = β1 (y)∼L2 , β2 (y)∼L2 , · · · , βL−1 (y)∼L2 , βL (y)∼L2 +L1
can be paired within η, where L2 could be very large compared with
L · 2L · (dim(X) + dim(Y ))3 , if the inductive limit system satisfies the very
slow dimension growth condition.
What we are going to prove is that, if SPφy and Θ(y) can be paired within η,
then they can still be paired within some small number (e.g., 2η), if one changes
a number—a small number compared with L—of maps βi to arbitrary maps
(in particular, to constant maps), provided that X is path connected and φ has
a certain spectral distribution property related to the number η and another
number δ (see 4.26 below). (Note that, how many maps are allowed to be
changed, also depends on η and δ.) (Those constant maps form the part of
the homomorphism defined by point evaluations.) At first sight, it might seem
impossible for this to be true. But, with the spectral distribution property of
the homomorphism φ, Lemma 2.15 of [Li2] (see Lemma 4.29 below) says that if
φ and another homomorphism ψ (in the application, ψ should be chosen to be
a homomorphism with the family of spectral functions Θ(y), i.e., SPψy = Θ(y)
for all y ∈ Y ) are close on the level of AffT , then their spectra SPφy and SPψy
can be paired within a small number. On the other hand, changing a very few
spectral functions (no matter how large a change in each function), will not
create a big change on the level of AffT (see 4.28 and the claims in 4.31 below).
Since the results of [Li 2] are not of a quantitative nature—they are for connecting homomorphisms φn,m with m large—, we can not apply them (2.18
and 2.19 of [Li 2]) directly. So we repeat part of the arguments in [Li 2].
The above method will lead us to Lemma 4.33 (see 4.26–4.33 for details). Then
our main decomposition theorems—Theorem 4.35 and Theorem 4.37— will be
more or less consequences.
Finally we remark that, in our decomposition, we cannot require that both
parts of the decomposition be homomorphisms as in 2.28 of [Li3], since in
general, C(X) is not stably generated (see [Lo]).
4.26. For the reader’s convenience, we will quote some notations, terminologies and results from [Li1] and [Li2].
Documenta Mathematica 7 (2002) 255–461
390
Guihua Gong
The following notation is inspired by a similar notation in [Li1].
For any η > 0, δ > 0, a homomorphism φ : P Mk (C(X))P → QMk′ (C(Y ))Q is
said to have the property sdp(η, δ) (spectral distribution property with
respect to η and δ) if for any η−ball
Bη (x) := {x′ ∈ X; dist(x′ , x) < η} ⊂ X
and any point y ∈ Y ,
#(SPφy ∩ Bη (x)) ≥ δ#(SPφy ),
counting multiplicity.
1
(Attension: The property sdp(r, δ) in [Li1] corresponds to sdp( 2r
, δ) above.)
Any homomorphism φ : ⊕ Mk (C(X)) → ⊕Ml (C(Y )) is said to have the
property sdp(η, δ) if each partial map has sdp(η, δ).
4.27. The following notations can be found in Section 2 of [Li2]. Let X be a
connected simplicial complex. For any closed set X1 ⊂ X, M > 0, let
if x ∈ X1
1
1
χX ,M (x) =
1
−
M
·
dist(x,
X
)
if dist(x, X1 ) ≤ M
1
1
1
0
if dist(x, X1 ) ≥ M .
For η > 0 and δ > 0, let
H1 (η) = {χX1 , 32
: X1 ⊂ X closed }.
η
Then there is a finite set H ⊂ H1 (η) such that for all h ∈ H1 (η), dist(h, H) < 8δ
(the distance is the distance defined by uniform norm). Denote such set by
H(η, δ, X) (⊂ C(X)). Although such a set is not unique, we fix one for each
triple (η, δ, X) for our purpose. (As pointed out in [Li1], the existence of such
finite set H(η, δ, X) follows from equi-continuity of the functions in H1 (η).)
4.28. For a unital C ∗ -algebra A, let TA denote the space of all tracial states
| , with
of A, i.e., τ ∈ T A if and only if τ is a positive linear map from A to C
τ (xy) = τ (yx) and τ (1) = 1. AffT A is the collection of all the affine maps from
T A to IR.
Any unital homomorphism φ : A → B induces an affine map
AffT φ : AffT A −→ AffT B.
It is well known, for any connected metrizable space X and any projection
P ∈ Mk (C(X)),
AffT (P Mk (C(X))P ) = AffT (C(X)) = CIR (X).
We would like to quote some easy facts about the AffT map from [Li1] and
[Li2].
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
391
If φ : C(X) → P Ml (C(Y ))P is a unital homomorphism and rank(P ) = k, then
AffT φ : C(X) → C(Y ) is given by
AffT φ(f ) =
l
1X
φ(f )ii ,
k i=1
where each φ(f )ii is the diagonal entry of φ(f ) ∈ P Ml (C(Y ))P ⊂ Ml (C(Y ))
at the place (i, i)
For a continuous map β : Y → X, let β ∗ : C(X) → C(Y ) be defined by
β ∗ (f ) = f ◦ β (∈ C(Y )) for any f ∈ C(X).
Suppose that β1 , β2 , · · · βl : Y → X are continuous maps. If ψ : C(X) →
Ml (C(Y )) is a homomorphism with {βi }li=1 as the set of spectral functions,
(e.g., ψ is defined by ψ(f ) = diag(β1∗ (f ), β2∗ (f ), · · · , βl∗ (f )),) then
l
AffT ψ(f ) =
1X ∗
β (f ).
l i=1 i
(Let H ⊂ CIR (X) be a finite subset satisfying kf k ≤ 1 for any f ∈ H. If one
modifies the above homomorphism ψ to a new homomorphism ψ ′ , by replacing
k functions from the set of spectral functions {βi }li=1 by other functions (from
Y to X), then
k
kAffT ψ(f ) − AffT ψ ′ (f )k ≤ , ∀f ∈ H.
l
′
In particular, this modification (from ψ to ψ ) does not create a big change on
the level of AffT , provided that k is very small compared with l, as mentioned
in 4.25.)
For a unital homomorphism φ : C(X) → P Ml (C(Y ))P with rank(P ) = k,
quoting from 1.9 of [Li1], we have
AffT φ(f )(y) =
1
k
X
f (xi (y)).
xi (y)∈SPφy
∼ Mrank(P ) (C
| )), which is
| )P (y) (=
Consider ey : P Ml (C(Y ))P → P (y)Ml (C
the homomorphism defined by evaluation at the point y. Then from the above
paragraph, we know that AffT (ey ◦ φ) depends only on SPφy . We can denote
ey ◦ φ by φ|y ( this is the homomorphism φ|{y} in 1.2.13 for the single point set
{y}).
Lemma 4.29.
([Li2,2.15]) Suppose that two unital homomorphisms
φ : C(X) → P Mk (C(Y ))P and ψ : C(X) → QMk (C(Y ))Q with rank(P ) =
rank(Q), satisfy the following two conditions:
η
, δ);
(1) φ has the property sdp( 32
(2) kAffT φ(h) − AffT ψ(h)k < 4δ , for all h ∈ H(η, δ, X).
Documenta Mathematica 7 (2002) 255–461
392
Guihua Gong
Then SPφy and SPψ y can be paired within η4 for any y ∈ Y .
(Notice that, no matter how small the δ is , the above conditions (1) and (2)
do not imply the other assumption rank(P ) = rank(Q), which is necessary for
our conclusion.)
η
Proof: If P = Q, this is exactly 2.15 of [Li2]. (Notice that, we use 32
in place of
η
of
[Li2
2.15],
so
our
conclusion
is
that,
SPφ
and
SPψ
can
be
paired
within
y
y
8
η
(instead
of
η).
Also
notice
that
the
set
H
in
[Li2
2.15]
is
chosen
to
be
the
4
same as the above set H(η, δ, X).)
To see the general case, fix y ∈ Y . We can consider two maps φ|y and ψ|y which
are unital homomorphisms from C(X) to C ∗ -algebras which are isomorphic to
| ). (Note that rank(P ) = rank(Q).) The
the same C ∗ -algebra Mrank(P ) (C
conditions (1) and (2) above imply the same conditions for φ|y and ψ|y , since
AffT φ(h)(y) = AffT (φ|y )(h). Therefore, by 2.15 of [Li 2], SPφy = SP (φ|y ) and
SPψ y = SP (ψ|y ) can be paired within η4 .
(If one checks the proof of 2.15 of [Li2] carefully, then he will easily recognize
that the above Lemma is already proved there.)
⊔
⊓
4.30. In the following paragraphs (4.30—4.32), we will apply the materials
from 2.8 – 2.10 of [Li2].
For any η > 0 and δ > 0, from 2.9 of [Li2], there exist a continuous map
α : [0, 1] → X, and a unital positive linear map ξ : C[0, 1] → C(X) such that
kξ ◦α∗ (f ) − f k <
δ
,
16
for each f ∈ H(η, δ, X), where α∗ : C(X) → C[0, 1] is induced by α. Furtherη
-dense in X.
more, we can choose α such that image(α) is 32
For α : [0, 1] → X, there is a σ > 0 such that |t − t′ | < 2σ implies that
dist(α(t), α(t′ )) <
η
.
32
For a fixed space X, the number σ depends only on η and δ, since so does the
continuous map α. We denote the number σ by σ(η, δ).
δ
> 0,
4.31. Let H̃ = α∗ (H(η, δ, X)) ⊂ C[0, 1]. For the finite set H̃ and 16
there is an integer N (as in Theorem 2.1 of [Li2]) such that for any positive
linear map ζ : C[0, 1] → C(Y ), and for any r ≥ N , there are r continuous maps
β1 , β2 , · · · , βr : Y −→ [0, 1]
such that
°
°
r
°
1X ∗ °
δ
°
°
β (f )° <
°ζ(f ) −
°
° 16
r i=1 i
for all f ∈ H̃, where βi∗ : C[0, 1] → C(Y ) is induced by βi .
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
393
δ
We will also assume N1 < 64
. Then we can prove the following claim.
Claim 1: For any r ≥ N , if
°
°
r
°
1X ∗ °
δ
°
°
for all f ∈ H̃,
βi (f )° <
°ζ(f ) −
°
° 16
r
i=1
then for any other two continuous maps τ1 , τ2 : Y −→ [0, 1],
°
!°
à r
° δ
°
X
1
°
°
∗
∗
∗
βi (f ) + τ1 (f ) + τ2 (f ) ° < , ∀f ∈ H̃
°ζ(f ) −
° 8
°
r+2
i=1
Proof of the claim: The claim follows from
°
à r
!°
°
°
X
1
°
°
∗
∗
∗
βi (f ) + τ1 (f ) + τ2 (f ) °
°ζ(f ) −
°
°
r + 2 i=1
°
° ° r
°
r
r
°
°
1X ∗ °
1 X ∗ °
°
° °1 X ∗
°
≤ °ζ(f ) −
β (f )° + °
βi (f ) −
βi (f )°
° °r
°
°
r i=1 i
r
+
2
i=1
i=1
°
°
° 1
°
∗
∗
°
+°
(τ
(f
)
+
τ
(f
))
2
°r + 2 1
°
δ
δ
δ
δ
<
+2·
+2·
=
16
64
64
8
for any f ∈ H̃. In the above estimation, we use the facts that kf k ≤ 1,
kβi∗ (f )k ≤ 1 and kτi∗ (f )k ≤ 1 for any f ∈ H̃.
In the above claim, if we replace the condition r ≥ N by the condition r ≥ mN , then in the conclusion, we can allow 2m continuous maps
τ1 , τ2 , · · · , τ2m : Y −→ [0, 1], instead of two maps. Namely, the following
claim can be proved in exactly the same way.
Claim 2: For any r ≥ mN , if
°
°
r
°
1X ∗ °
δ
°
°
,
∀ f ∈ H̃,
β (f )° <
°ζ(f ) −
°
° 16
r i=1 i
then for any 2m continuous maps τ1 , τ2 , · · · , τ2m : Y −→ [0, 1],
°
à r
!°
2m
° δ
°
X
X
1
°
°
∗
∗
τi (f ) ° < , ∀f ∈ H̃.
βi (f ) +
°ζ(f ) −
° 8
°
r + 2m
i=1
³
i=1
´
1
+ 1, where int(·) denote the integer part of the
4.32. Let n = int σ(η,δ)
number (see 1.1.7 (c)).
Divide [0, 1] into n intervals such that each of them has length at most σ(η, δ).
Choose n points
t1 , t 2 , · · · , t n ,
Documenta Mathematica 7 (2002) 255–461
394
Guihua Gong
one from each of the intervals. Let xi = α(ti ) ∈ X, i = 1, 2, · · · , m. Then the
set
{x1 , x2 , · · · , xn }
η
is 16
-dense in X by the way σ is chosen in 4.30.
From the above discussion, for fixed η > 0, δ > 0, and the space X, we can find
η
-dense set
α, ξ, σ, N, n, H(η, δ, X), H̃, the set {t1 , t2 , · · · , tn } ⊂ [0, 1], and the 16
{x1 , x2 , · · · , xn } ⊂ X. All of them depend only on η, δ, and the space X.
Lemma 4.33. For any connected simplicial complex X, any numbers η > 0 and
δ > 0, there are integers n, N , a continuous map α : [0, 1] → X, and finitely
η
many points {t1 , t2 , · · · , tn } ⊂ [0, 1] with {α(t1 ), α(t2 ), · · · , α(tn )} 16
-dense in
X, such that the following is true. (Denote L := n(N + 2).)
If a unital homomorphism φ : C(X) → P Mk (C(Y ))P satisfies the following
two conditions:
η
, δ);
(i) φ has the property sdp( 32
2
(ii) rankφ(1) := K ≥ L = (n(N + 2))2 ,
and write K = LL2 +L1 with L2 = int( K
L ) and 0 ≤ L1 < L, (note that L ≤ L2 ,
since K ≥ L2 ,)
then there are L continuous functions
β1 , β2 , · · · , βn , βn+1 , · · · , βL : Y −→ [0, 1]
such that
(1) βi (y) = ti for 1 ≤ i ≤ n;
(2) For each y ∈ Y, SPφy and the set
ª
©
Θ(y) = α◦β1 (y)∼L2 , α◦β2 (y)∼L2 , · · · , α◦βL−1 (y)∼L2 , α◦βL (y)∼L2 +L1
can be paired within η2 .
(3) If Y is a connected finite simplicial complex and Y 6= {pt}, then the map
βn+1 : Y → [0, 1]—the first nonconstant map above—, is a surjection.
(This lemma is similar to Lemma 2.18 of [Li2], but we require some of the
functions βi (1 ≤ i ≤ n) to be constant functions.)
(Attention: To apply Theorem 4.1, one only needs SPφy and Θ(y) to be
paired within η. The advantage of using η2 is the following. If ψ is another
homomorphism such that SPψy and SPφy can be paired within η2 for any y,
then we can apply Theorem 4.1 to both φ and ψ without requiring ψ to have
η
, δ). This observation will not be used in the proof of the
the property sdp( 32
main theorem of this paper. But it will be used in the proof of the Uniqueness
Theorem in [EGL] (part II of the series), see 4.41–4.48 below.)
Proof: Follow the notations in 4.26 – 4.32. Let
ζ : C[0, 1] −→ C(Y )
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
395
be defined by ζ = AffT φ◦ξ. Since K − 2nL2 ≥ nL2 N, there are K − 2nL2
continuous maps
γ1 , γ2 , · · · , γK −2nL2 : Y −→ [0, 1]
such that
for all f ∈ H̃. Let
°
°
1
°
°ζ(f ) −
°
K − 2nL2
K −2nL2
X
i=1
°
°
°
γi∗ (f )°
°
<
δ
16
β1 , β2 , · · · , βn : Y −→ [0, 1]
be defined by βi (y) = ti . Then by Claim 2 of 4.31 (taking m = nL2 ),
°
ÃK −2nL
!°
n
°
° δ
X2
X
1
°
°
∗
∗
γi (f ) + 2L2
βi (f ) ° <
°ζ(f ) −
°
° 8
K
i=1
i=1
for all f ∈ H̃ ⊂ C[0, 1]. Therefore,
°
ÃK −2nL
!°
n
°
° δ
X
X2
1
°
°
∗
∗
∗
(α◦γi ) (f ) + 2L2
(α◦βi ) (f ) ° <
°(ζ ◦α )(f ) −
°
° 8
K
i=1
i=1
for all f ∈ H(η, δ, X). On the other hand, by 4.30 and ζ = AffT φ◦ξ,
kAffT φ(f ) − (ζ ◦α∗ )(f )k <
δ
16
for f ∈ H(η, δ, X).
One can define a unital homomorphism ψ : C(X) → MK (C(Y )) with
2
∪ {(α◦βi )∼2L2 }ni=1 as the family of the spectral functions. Then
{α◦γi }K−2nL
i=1
from 4.28,
ÃK −2nL
!
n
X2
X
1
∗
∗
(α◦γi ) (f ) + 2L2
(α◦βi ) (f ) .
AffT ψ(f ) =
K
i=1
i=1
Hence,
δ
δ
δ
+
<
8 16
4
for all f ∈ H(η, δ, X). Note that rank(P ) = K. By Lemma 4.29, SPφy and
kAffT φ(f ) − AffT ψ(f )k ≤
SPψ y =
ª
©
α◦β1 (y)∼2L2 , α◦β2 (y)∼2L2 , · · · , α◦βn (y)∼2L2 , α◦γ1 (y), · · · , α◦γL−2nL2 (y)
can be paired within η4 .
Note that in our lemma, we only need L2 copies of each constant maps βi
(i = 1, 2, · · · , n). One may wonder why we put 2L2 copies of each of maps βi
in the above set. The reason is that, after taking out L2 copies of βi , we still
want the set Θ(y) to have enough elements in each small interval of length σ,
and the other L2 copies of βi can serve for this purpose.
Documenta Mathematica 7 (2002) 255–461
396
Guihua Gong
Consider the following set (of K − nL2 elements)
{β1∼L2 (y), β2∼L2 (y), · · · , βn∼L2 (y), γ1 (y), γ2 (y), · · · , γK −2nL2 (y)}.
In each interval of [0, 1] of length σ, there are at least L2 points (counting
multiplicities) in the above set.
The following argument appeared in 2.18 of [Li2].
For each fixed y, we can rearrange all the elements in the above set in the
′
increasing order. I.e., write them as γ1′ (y), γ2′ (y), · · · , γK
−nL2 (y) such that for
each fixed y
′
{γ1′ (y), γ2′ (y), · · · , γK
−nL2 (y)} =
= {β1 (y)∼L2 , β2 (y)∼L2 , · · · , βn (y)∼L2 , γ1 (y), γ2 (y), · · · , γK −2nL2 (y)}
(as a set with multiplicity), and such that
′
0 ≤ γ1′ (y) ≤ γ2′ (y) ≤ · · · ≤ γK
−nL2 (y) ≤ 1.
It is easy to prove that γi′ (y), 1 ≤ i ≤ K − nL2 are continuous (real-valued)
functions, using the following well known fact repeatedly: For any two realvalued continuous functions f and g, the functions max(f, g) and min(f, g) are
also continuous.
We can put each group of L2 consecutive functions of {γi′ } (beginning with
smallest one) together except the last L2 + L1 functions which will be put into
a single group—the last group. Then we replace all the functions in a same
group by the smallest function in the group. Namely, let
βn+1 = γ1′ , βn+2 = γL′ 2 +1 , · · · , βL = γ(′L−n−1)L2 +1 .
Then from the fact that in each interval of [0, 1] of length σ, there are at least
′
L2 points (counting multiplicity) in the set {γ1′ (y), γ2′ (y), · · · , γK
−nL2 (y)}, we
′
′
′
know that {γ1 (y), γ2 (y), · · · , γK −nL2 (y)} and
{βn+1 (y)∼L2 , βn+2 (y)∼L2 , · · · , βL−1 (y)∼L2 , βL (y)∼L2 +L1 }
can be paired within 2σ. Recall that |t−t′ | < 2σ implies that dist(α(t), α(t′ )) <
η
16 . Hence
©
α◦β1 (y)∼2L2 , α◦β2 (y)∼2L2 , · · · , α◦βn (y)∼2L2 ,
α◦γ1 (y), α◦γ2 (y), · · · , α◦γL−2nL2 (y)}
and
©
Θ(y) = α◦β1 (y)∼L2 , α◦β2 (y)∼L2 , · · · , α◦βn (y)∼L2 ,
α◦βn+1 (y)∼L2 , · · · , α◦βL−1 (y)∼L2 , α◦βL (y)∼L2 +L1
Documenta Mathematica 7 (2002) 255–461
ª
Simple Inductive Limit C ∗ -Algebras, I
397
η
can be paired within 16
. Therefore, SPφy and Θ(y) can be paired within
η
5η
η
4 + 16 < 16 .
η
Note that {α◦β1 (y), α◦β2 (y), · · · , α◦βn (y)} is 16
dense in X. From the proof
of Lemma 1.2.17, if we replace only one map (say βn+1 ) by an arbitrary map
from Y to [0, 1], then the new Θ(y) can be paired with the old Θ(y) to within
η
8 . As a consequence, we still have that SPφy and the new Θ(y) can be paired
η
η
within 5η
16 + 8 < 2 . In particular, if Y is a connected finite simplicial complex
which is not a single point, then βn+1 could be chosen to be a surjection, again
using the Peano curve.
⊔
⊓
4.34. Fix a large positive integer J. We require that the decomposition in
4.1 to satisfy the condition
J · (rank(p0 ) + 2 dim(Y )) ≤ rank(pi ), ∀i ≥ 1.
To do so, we need rank(φ(1)) to be large enough. We describe it as follows.
For a connected simplicial complex X, and for numbers η > 0 and δ > 0, let
N, n and α : [0, 1] → X be as in Lemma 4.33. Let L = n(N + 2). Suppose that
η
, δ)
φ : C(X) → Mk (C(Y )) is a homomorphism. If φ has the property sdp( 32
and
rank(φ(1)) ≥ 2JL2 · 2L (dim X + dim Y + 1)3 ,
then there are continuous functions
β1 , β2 , · · · , βL : Y −→ [0, 1]
(as in Lemma 4.33) such that SPφy and the set
©
ª
α◦β1 (y)∼L2 , α◦β2 (y)∼L2 , · · · , α◦βL−1 (y)∼L2 , α◦βL (y)∼L2 +L1
can be paired within η2 , where
¶
µ
rankφ(1)
≥ 2JL · 2L (dim X + dim Y + 1)3 ,
L2 = int
L
and 0 ≤ L1 < L. For any given set F ⊂ C(X), if η is chosen as in Theorem
4.1 (see 4.4), then by TheoremP
4.1, there are mutually orthogonal projections
L
p1 , p2 , · · · , pL and p0 = φ(1) − i=1 pi such that
(1) For all f ∈ F and y ∈ Y ,
kφ(f )(y) − p0 (y)φ(f )(y)p0 (y) ⊕
L
M
f (ai (y))pi (y)k < ε;
i=1
(2) For each i = 1, 2, · · · , L, rank(pi ) ≥ L2 − 2L (dim X + dim Y + 1)3 , and
J(rank(p0 ) + 2 dim(Y )) ≤ J(L · 2L (dim X + dim Y + 1)3 + 2 dim Y ) ≤ rank(pi ).
Documenta Mathematica 7 (2002) 255–461
398
Guihua Gong
By [Hu], p0 ⊕ p0 ⊕ · · · ⊕ p0 is (unitarily) equivalent to a subprojection of pi ,
{z
}
|
J
since every complex vector bundle (over Y ) of dimension J · rank(p0 ) is a subbundle of any vector bundle (over Y ) of dimension at least J ·rank(p0 )+dim(Y ).
We denote this fact by J[p0 ] < [pi ].
Let Q0 = p0 , Q1 = p1 + p2 + · · · + pn and Q2 = pn+1 + pn+2 + · · · + pL . Then
φ(1) = Q0 + Q1 + Q2 .
Let φ0 : C(X) → Q0 Mk (C(Y ))Q0 , φ1 : C(X) → Q1 Mk (C(Y ))Q1 and φ2 :
C(X) → Q2 Mk (C(Y ))Q2 be defined by
φ0 (f )(y) = p0 φ(f )(y)p0 ,
φ1 (f ) =
n
X
f (α◦βi (y))pi ,
and
i=1
φ2 (f )(y) =
L
X
f (α◦βi (y))pi .
i=n+1
Then we have the following facts.
(a) φ2 is a homomorphism factoring through C[0, 1] as
ξ1
ξ2
φ2 : C(X) −→ C[0, 1] −→ Q2 Mk (C(Y ))Q2 .
Furthermore, if Y 6= {pt}, then ξ2 is injective. (This follows from the surjection
of βn+1 .)
(b) Note that
α◦β1 (y) = x1 , α◦β2 (y) = x2 , · · · , α◦βn (y) = xn
are n constant maps with {x1 , x2 , · · · , xn } η-dense in X. By the claim in 4.20,
(φ0 ⊕ φ1 )(F ) is approximately constant to within ε. (Note that φ0 is not a
homomorphism, it is a completely positive linear ∗-contraction.)
Furthermore, if η < ε, then the set {x1 , x2 , · · · , xn } is ε-dense in X.
Therefore, we have proved the following theorem.
Theorem 4.35. Let X be a connected finite simplicial complex, and ε > η > 0.
For any δ > 0, there is an integer L > 0 such that the following holds.
Suppose that F ⊂ C(X) is a finite set such that dist(x, x′ ) < 2η implies |f (x) −
f (x′ )| < 3ε for all f ∈ F .
η
, δ),
If φ : C(X) → Mk (C(Y )) is a homomorphism with the property sdp( 32
and rank(φ(1)) ≥ 2J · L2 · 2L (dim X + dim Y + 1)3 , where Y is a connected finite simplicial complex and J is any fixed positive integer, then there
are three mutually orthogonal projections Q0 , Q1 , Q2 ∈ Mk (C(Y )), a map
φ0 ∈ Map(C(X), Q0 Mk (C(Y ))Q0 )1 and two homomorphisms
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
399
φ1 ∈ Hom(C(X), Q1 Mk (C(Y ))Q1 )1 and φ2 ∈ Hom(C(X), Q2 Mk (C(Y ))Q2 )1
such that
(1) φ(1) = Q0 + Q1 + Q2 ;
(2) kφ(f ) − φ0 (f ) ⊕ φ1 (f ) ⊕ φ2 (f )k < ε for all f ∈ F ;
(3) The homomorphism φ2 factors through C[0, 1] as
ξ2
ξ1
φ2 : C(X) −→ C[0, 1] −→ Q2 Mk (C(Y ))Q2 .
Furthermore, if Y 6= {pt}, then ξ2 is injective;
(4) The set (φ0 ⊕ φ1 )(F ) is approximately constant to within ε;
(5) Q1 = p1 + · · · + pn , with J[Q0 ] ≤ [pi ] (i = 1, 2 · · · n), φ0 is defined by
φ0 (f ) = Q0 φ(f )Q0 , and φ1 is defined by
φ1 (f ) =
n
X
i=1
f (xi )pi , ∀f ∈ C(X),
where p0 , p1 , · · · pn are mutually orthogonal projections and {x1 , x2 , · · · xn } ⊂ X
is an ε-dense subset of X. (Again by J[p] ≤ [q], we mean that p ⊕ p ⊕ · · · ⊕ p
{z
}
|
J
is (unitarily) equivalent to a subprojection of q.)
Furthermore, we can choose any two of projections Q0 , Q1 , Q2 to be trivial, if we
wish. If φ(1) is trivial, then all of them can be chosen to be trivial projections.
(This is remark 4.22.)
4.36.
Let a simple C ∗ -algebra ALbe an inductive limit of matrix algebras
tn
over simplicial complexes (An =
i=1 M[n,i] (C(Xn,i )), φn,m ) with injective
homomorphisms. Suppose that this inductive limit system possesses the very
slow dimension growth condition.
In what follows, we will L
use the material from 1.2.19.
tn
ε
Fix An , finite set Fn = i=1
Fni ⊂ An , and ε > 0. Let ε′ = max1≤i≤t
{[n,i]} .
n
i
Let F ′ ⊂ C(Xn,i ) be the finite set consisting of all the entries of elements in
Fni (⊂ M[n,i] (C(Xn,i ))). Let η > 0 (η ≤ ε) be such that if x, x′ ∈ Xn,i (i =
′
i
1, 2 · · · tn ) and dist(x, x′ ) < 2η, then |f (x) − f (x′ )| < ε3 for any f ∈ F ′ .
For the above η > 0, there is a δ > 0 such that for sufficiently large m,
η
i
j
each partial map φi,j
n,m : An → Am has the property sdp( 32 , δ). (This is a
consequence of simplicity of the algebra A and injectivity of φn,m . See [DNNP],
[Ell], [Li1-2] for details.)
For these numbers η and δ, and the simplicial complexes Xn,i , there are
L(i), i = 1, 2, · · · , tn , as in Theorem 4.35. (Note that the numbers Li only
depend on η, δ and the spaces.) Let L = maxi L(i). Fix a positive integer J.
By the very slow dimension growth condition, there is an integer M such that
for any m ≥ M ,
rankφi,j
n,m (1Ain )
> 2J · L2 · 2L (dim Xn,i + dim Xm,j + 1)3 .
rank(1Ain )
Documenta Mathematica 7 (2002) 255–461
400
Guihua Gong
As in 1.2.16, (also see 1.2.19) each partial map
′
j i,j
i,j
i
φi,j
n,m : An → φn,m (1Ain )Am φn,m (1Ain ) can be written as φ ⊗ 1[n,k] for some
homomorphism φ′ : C(Xn,i ) → EAjm E, where E = φi,j
n,m (e11 ), and e11 is the
canonical matrix unit corresponding to the upper left corner. The map φ′ also
η
, δ).
has the property sdp( 32
i
Applying Theorem 4.35 to F ′ ⊂ C(Xn,i ), η, δ, and φ′ (as the above) and using
1.2.19, one can obtain the following Theorem.
Ltn
i
Theorem 4.37. For any An , finite set F =
i=1 F ⊂ An , positive integer J, and number ε > 0, there are an Am , mutually orthogonal projections Q0 , Q1 , Q2 ∈ Am with Q0 + Q1 + Q2 = φn,m (1An ), a unital map ψ0 ∈
Map(An , Q0 Am Q0 )1 , and unital homomorphisms ψ1 ∈ Hom(An , Q1 Am Q1 )1 ,
ψ2 ∈ Hom(An , Q2 Am Q2 )1 , such that
(1) kφn,m (f ) − ψ0 (f ) ⊕ ψ1 (f ) ⊕ ψ2 (f )k < ε for all f ∈ F ;
(2) The set (ψ0 ⊕ ψ1 )(F ) is weakly approximately
Ltn constant to within ε;
(3) The homomorphism ψ2 factors through i=1
M[n,i] (C[0, 1]) as
ξ1
ψ2 : An −→
tn
M
i=1
ξ2
M[n,i] (C[0, 1]) −→ Q2 Am Q2 ,
and ξ2 satisfies the following condition: if Xm,j 6= {pt}, then ξ2i,j :
M[n,i] (C[0, 1]) → Ajm is injective;
i,j
i,j
i,j
j
(4) Each partial map ψ0i,j : Ain → Qi,j
0 Am Q0 (where Q0 = ψ0 (1Ain )) is
of the form ψ0′ ⊗ id[n,k] with ψ0′ : C(Xn,i ) → q0 Ajm q0 (where q0 = ψ0i,j (e11 )
i,j
i,j
j
is a projection). Each partial map ψ1i,j : Ain → Qi,j
1 Am Q1 (where Q1 =
i,j
i
i,j j i,j
′
′
ψ1 (1Ain )) is of the form ψ1 ⊗ id[n,k] and ψ1 : An → p Am p (where pi,j =
ψ1i,j (e11 )), satisfies the following
ψ1′ (f ) =
n
X
f (xi )pi
i=1
for any f ∈ C(Xn,i ), where p1 , · · · , pn are mutually orthogonal projections with
pi,j = p1 +· · ·+pn , and with J ·[q0 ] ≤ [ps ] (s = 1, 2 · · · n) and {x1 , x2 , · · · xn } ⊂
Xn,i is an ε-dense subset in Xn,i .
(When we apply this theorem in Section 6, Q0 + Q1 will be chosen to be a
trivial projection.)
Definition 4.38. Let A = P Ml (C(X))P , and L be a positive integer and
η > 0. A homomorphism λ : A → B = QMl1 (C(Y ))Q is said to be defined
by point evaluations of size at least L at an η-dense subset if there
are mutually orthogonal projections Q1 , Q2 , · · · , Qn with rank(Qi ) ≥ L, an
η-dense subset {x1 , x2 , · · · , xn } ⊂ X, and unital homomorphisms λi : A →
Qi BQi , i =
2, · · · , n such that
P1,
Ln
n
(1) λ(1) = i=1 Qi , and λ = i=1 λi ;
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
401
∼M
|
| )P (xi ) (=
(2) The homomorphisms λi factor through P (xi )Ml (C
rank(P ) (C))
as
exi
λ′i
| )P (xi )−→Q
(xi )Ml (C
λi = λ′i ◦ exi : P Ml (C(X))P −→P
i BQi ,
where exi are evaluation maps defined by exi (f ) = f (xi ) and
| ), Qi BQi )1 .
λ′i ∈ Hom(Mrank(P ) (C
We will also call the above homomorphism λ to have the property PE(L, η).
(PE stands for point evaluation.)
A homomorphism λ : A → B = QMl1 (C(Y ))Q is said to contain a part of
point evaluation at point x of size at least L, if λ = λ1 ⊕ λ′ , where λ1
| )P (x) as
factor through P (x)Ml (C
λ′
e
1
x
| )P (x)−→Q
(x)Ml (C
λ1 = λ′1 ◦ ex : P Ml (C(X))P −→P
1 BQ1 ,
and λ′1 is a unital homomorphism with rank(Q1 ) ≥ L.
The following result is a corollary of Theorem 4.37, and will also be used in the
proof of our main reduction theorem.
Ltn
i
Corollary 4.39.
For any An , finite set F =
i=1 F ⊂ An , positive
integer J, any numbers ε > 0 and η > 0, and any projection P = ⊕P i ∈
⊕Ain , there are an Am , mutually orthogonal projections Q0 , Q1 , Q2 ∈ Am with
Q0 +Q1 +Q2 = φn,m (1An ), a unital map ψ0 ∈ Map(An , Q0 Am Q0 )1 , and unital
homomorphisms ψ1 ∈ Hom(An , Q1 Am Q1 )1 , ψ2 ∈ Hom(An , Q2 Am Q2 )1 , such
that
Part I:
(1) kφn,m (f ) − ψ0 (f ) ⊕ ψ1 (f ) ⊕ ψ2 (f )k < ε for all f ∈ F ;
(2) The homomorphism ψ2 factors through a direct sum of matrix algebras over
C[0, 1] as
tn
ξ2
ξ1 M
M[n,i] (C[0, 1]) −→ Q2 Am Q2 ,
ψ2 : An −→
i=1
and ξ2 satisfies the condition that, if Xm,j 6= {pt}, then ξ2i,j : M[n,i] (C[0, 1]) →
Ajm is injective.
(3) For any blocks Ain ⊂ An , Ajm ⊂ Am , and for the partial maps ψ0i,j and
i,j
ψ1i,j , we have that ψ0i,j (1Ain ) := Qi,j
0 is a projection and ψ1 has the property
PE(J · rank(Qi,j
0 ), η).
(4) The set (ψ0 ⊕ ψ1 )(F ) is weakly approximately constant to within ε.
Part II:
ψ0i,j (P i ) and ψ0i,j (1Ain − P i ) are mutually orthogonal projections, and the decomposition of φ′n,m := φn,m |P An P as the direct sum of ψ0′ := ψ0 |P An P ,
ψ1′ := ψ1 |P An P , and ψ2′ := ψ2 |P An P satisfies the following conditions:
(1) kφ′n,m (f ) − ψ0′ (f ) ⊕ ψ1′ (f ) ⊕ ψ2′ (f )k < ε for all f ∈ P F P = ⊕P i F i P i ;
(2) The homomorphism ψ2′ factors through a C ∗ -algebra C which is a direct
sum of matrix algebras over C[0, 1] as
ξ′
ξ′
1
2
ψ2′ : P An P −→
C −→
Q′2 Am Q′2 ,
Documenta Mathematica 7 (2002) 255–461
402
Guihua Gong
and ξ2′ satisfies the following condition, if Xm,j 6= {pt}, then ξ2i,j : C i → Ajm is
injective, where Q′2 = ψ2 (P ).
′
(3) For any blocks Ain ⊂ An , Ajm ⊂ Am , and for the partial maps ψ0i,j and
′
′
′
′
ψ1i,j , we have that ψ0i,j (P i ) := Q0i,j is a projection and ψ1i,j has property
′
PE(J · rank(Q0i,j ), η).
Proof: Obviously, the first part of the corollary follows from Theorem 4.37.
To prove the second part, we only need to perturb ψ0 ∈ Map(A, Q0 BQ0 )1 to
newψ0 such that the restriction newψ0 |D is a homomorphism, where
M
M
| · Pi ⊕
| · (1 i − P i )
D :=
C
C
An
i
i
is a finite dimensional subalgebra of An .
By Lemma 1.6.8, such perturbation exists if ψ0 is sufficiently multiplicative,
which is automatically true if the set F is large enough and the number ε is
small enough, using theLnext lemma.
tn
M[n,i] (C[0, 1]))ξ1 (P ) is still a direct sum of matrix
(Note that C := ξ1 (P )( i=1
algebras over C[0, 1], since all the projections in M• (C[0, 1]) are trivial.)
⊔
⊓
Lemma 4.40. Let A be a unital C ∗ -algebra. Suppose that G ⊂ A is a finite set
containing 1A , and G1 = G × G := {gh | g ∈ G, h ∈ G}. Suppose that δ > 0,
1
and δ ′ = 31 kGk
δ, where kGk = maxg∈G {kgk}.
Suppose that B is a unital C ∗ -algebra and p ∈ B is a projection. If a
homomorphism φ ∈ Hom(A, B) and two maps φ1 ∈ Map(A, pBp), φ2 ∈
Map(A, (1 − p)B(1 − p)) satisfy
kφ(g) − φ1 (g) ⊕ φ2 (g)k < δ ′ , ∀g ∈ G1 ,
then both φ1 and φ2 are G-δ multiplicative.
Proof: The proof is straight forward, we omit it.
Theorem 4.37 and Corollary 4.39 will be used in the proof of our Main Reduction Theorem in this article. Theorem 4.35 will be used in the proof of the
Uniqueness Theorem in [EGL]. The rest of this section will not be used in this
paper. They are important to [EGL].
4.41. In the rest of this section, we will compare the decompositions of two
different homomorphisms. Such comparison will be used in the proof of the
Uniqueness Theorem in [EGL].
Let X, η, δ, φ, {βi }Li=1 , and Θ(y) be as in 4.34. (Take J = 1.) Suppose that
φ : C(X) → Mk (C(Y )) is as in Theorem 4.35, and ψ : C(X) → Mk (C(Y )) is
another homomorphism with φ(1) = ψ(1). If
kAffT φ(f ) − AffT ψ(f )k <
δ
4
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
403
for all f ∈ H(η, δ, X), then by Lemma 4.29, SPφy and SPψ y can be paired
within η2 . Since SPφy and Θ(y) can be paired within η2 , SPψ y and Θ(y) can be
paired within η. Similar to 4.34, by Theorem 4.1, there are
PLmutually orthogonal
projections q1 , q2 , · · · , qn , qn+1 , · · · , qL and q0 = ψ(1) − i=1 qi such that
(1) For all y ∈ Y and f ∈ F ,
kψ(f )(y) − q0 ψ(f )(y)q0 ⊕
L
X
i=1
f (α◦βi (y))qi k < ε.
(2) rank(q0 ) + 2 dim(Y ) ≤ rank(qi ).
As Remark 4.22, we can choose projections pi for φ and qi for ψ to be trivial
projections with rank(pi ) = rank(qi ). (Note that, in 4.34, the number L2
and L2 + L1 , which serve as Ti , i = 1, 2, · · · , L (i.e., Ti = L2 , for 1 ≤ i ≤
L − 1, and TL = L2 + L1 ) in Theorem 4.1, are very larger.) Therefore, there
is a unitary u ∈ Mk (C(Y )) such that
uqi u∗ = pi ,
i = 1, 2, · · · , L.
Let ψ̃ = Adu◦ψ. Then
kψ̃(f )(y) − p0 ψ̃(f )(y)p0 ⊕
L
X
i=1
f (α◦βi (y))pi k < ε
for all y ∈ Y and f ∈ F .
Note that the above decomposition has P
the same form as that of φ, even with
L
the
same
projections
p
and
the
part
i
i=1 f (α◦βi (y))pi . Also, in the part
PL
f
(α
◦βi (y))pi , there is a map defined by point evaluations:
i=1
φ′ (f ) =
n
X
f (xi )pi ,
i=1
with {x1 , x2 , · · · , xn } η-dense in X, and rank(pi ) ≥ rank(p0 ) + 2 dim(Y ). This
means that two different homomorphisms which are close at the level of AffT
can be decomposed in the same way. This result will be useful in the proof of
the Uniqueness Theorem for certain spaces X with K1 (C(X)) a torsion group.
We summarize what we obtained as the following proposition which will be used
in the proof of the Uniqueness Theorem for certain spaces X with K1 (C(X))
a torsion group.
Proposition 4.42. Let X be a connected simplicial complex, ε > 0, and
F ⊂ C(X) be a finite set.
Suppose that η ∈ (0, ε) satisfies that if dist(x, x′ ) < 2η, then |f (x) − f (x′ )| < 3ε
for all f ∈ F .
For any δ > 0, there is an integer L > 0 and a finite set H ⊂ AffT (C(X))(=
C(X)) such that the following holds.
Documenta Mathematica 7 (2002) 255–461
404
Guihua Gong
If φ, ψ : C(X) → Mk (C(Y )) are homomorphisms with properties
η
(a) φ has sdp( 32
, δ);
(b) rank(φ(1)) ≥ 2L2 · 2L (dim X + dim Y + 1)3 ;
(c) φ(1) = ψ(1) and
kAffT φ(h) − AffT ψ(h)k <
δ
, ∀h ∈ H,
4
then there are two orthogonal projections Q0 , Q1 ∈ Mk (C(Y )), two
maps φ0 , ψ0 ∈ Map(C(X), Q0 Mk (C(Y ))Q0 )1 , a homomorphism φ1 ∈
Hom(C(X), Q1 Mk (C(Y ))Q1 )1 , and a unitary u ∈ Mk (C(Y )) such that
(1) φ(1) = ψ(1) = Q0 + Q1 ;
(2) kφ(f ) − φ0 (f ) ⊕ φ1 (f )k < ε, and k(Adu ◦ ψ)(f ) − ψ0 (f ) ⊕ φ1 (f )k < ε for
all f ∈ F ;
(3) φ1 factors through C[0, 1].
(4) Q0 = p0 + p1 + · · · + pn with rank(p0 ) + 2 dim(Y ) ≤ rank(pi ) (i = 1, 2 · · · n),
and φ0 and ψ0 are defined by
φ0 (f ) = p0 φ(f )p0 +
n
X
i=1
ψ0 (f ) = p0 (Adu ◦ ψ)(f )p0 +
f (xi )pi , ∀f ∈ C(X),
n
X
i=1
f (xi )pi , ∀f ∈ C(X),
where p0 , p1 , · · · pn are mutually orthogonal projections and {x1 , x2 , · · · xn } ⊂ X
is an ε-dense subset in X.
(Comparing with Theorem 4.35, the maps φ0 and φ1 in 4.35 have been put
together to form the map φ0 in the above proposition.)
(In [EGL], we will prove that the above φ0 and ψ0 are approximately unitarily
equivalent to each other to within some small number (under the condition
KK(φ) = KK(ψ)), then so also are φ and ψ.)
4.43. The above proposition is not strong enough to prove the Uniqueness
Theorem for homomorphisms from C(S 1 ) to Mk (C(Y )), since K1 (C(S 1 )) is
infinite. Before we conclude this section, we introduce a result which can be
used to deal with this case (i.e, the case S 1 ).
We will discuss briefly what the problem is, and how to solve the problem.
Suppose that φ and ψ are two homomorphisms from C(S 1 ) to another C ∗ algebra, For φ and ψ to be approximately unitarily equivalent to each other,
they should agree not only on AffT (C(S 1 )) and K∗ (C(S 1 )), but also on the
determinant functions. That is, φ(z)ψ(z)∗ should have only a small variation in
the determinant, where z ∈ C(S 1 ) is the standard generator. (All these things
will be made precise in [EGL].) This idea has appeared in [Ell2] and [NT].
Roughly speaking, if φ and ψ agree (approximately) to within ε at the level of
the determinant (this will also be made precise in [EGL]), then the maps p0 φp0
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
405
and p0 (Adu ◦ ψ)p0 from Proposition 4.42 agree only to within rank(φ(1)) ε at
rank(p0 )
the level of the determinant. So for the decomposition to be useful for the
proof of the uniqueness theorem, rank(p0 ) should not be too small compared
with rank(φ(1)). (This will be the property (2) of Theorem 4.45 below.) On
the other hand, in the decompositions of φ and Adu ◦ ψ, we also need the
homomorphism defined by point evaluations, which by Proposition 4.42 is the
same for both of these decompositions, to be large in order to absorb the parts
p0 φp0 and p0 (Adu ◦ ψ)p0 . This will be the property (3) of Theorem 4.45.
Therefore, rank(p0 ) should not be too large either.
η
To do that, besides the property sdp( 32
, δ), we also need sdp property for an
η̃
extra pair ( 32 , δ̃), where η̃ depends on δ. For those readers who are familiar with
[Ell2] and [NT], we encourage them to compare the sdp property for the two
η̃
η
pairs ( 32
, δ) and ( 32
, δ̃), with the conditions of Theorem 4 of [Ell2], and Lemma
2.3 and Theorem 2.4 of [NT], in the following way. In Theorem 4 of [Ell2] (see
page 100 of [Ell2]), roughly speaking, the sentence on lines 16–20 corresponds
η̃
, δ̃), and the sentence on lines 21–22 corresponds to our
to our property sdp( 32
η
η
η
1
property sdp( 32 , δ). That is, m
corresponds to our 32
(or 16
in some sense),
η̃
1
3
corresponds
to
our
δ,
corresponds
to
our
,
and
δ
corresponds
to our δ̃.
n
n
32
η
Similarly, in Lemma 2.3 of [NT], condition (1) corresponds to our sdp( 32
, δ)
η̃
and condition (2) corresponds to our sdp( 32 , δ̃). Also in Theorem 2.4 of [NT],
η
condition (2) corresponds to our sdp( 32
, δ) and condition (3) corresponds to
η̃
our sdp( 32
, δ̃).
Such a construction will be given in 4.44 below.
In 4.44, we will first describe the condition that φ should satisfy. Then we will
carry out the construction in three steps.
In Step 1, we will follow the procedure in 4.34, to decompose φ into p0 φp0 ⊕ φ1 ,
η̃
η
corresponding to the property sdp( 32
, δ̃) (not sdp( 32
, δ)). (Here, the map φ1
is φ1 ⊕ φ2 in the notation of 4.34 or 4.35.)
In Step 2, we will take a part p′ φ1 p′ out of φ1 and add it to p0 φp0 to obtain
P0 φP0 , where P0 = p0 + p′ . The rest of φ1 will be defined to be newφ1 . In this
way, we can get the projection P0 with suitable size (neither too small nor too
large). The size depends on δ, which explains why η̃ depends on δ.
In Step 3. we will prove that newφ1 can be decomposed again in such a way
that the point evaluation part of its decomposition is sufficiently large that it
can be used to control P0 φP0 , in the proof of the uniqueness theorem in [EGL].
η
(See the property (3) of Theorem 4.45.) The property sdp( 32
, δ) is used in this
step.
4.44. Let F ⊂ C(X) be a finite set, ε > 0 and ε1 > 0. Suppose that the
positive number η < ε41 satisfies the condition that, if dist(x, x′ ) < 2η, then
kf (x) − f (x′ )k <
ε
.
3
For any δ > 0, consider the pair (η, δ) as in 4.33. Let N, n be as in 4.33. Instead
Documenta Mathematica 7 (2002) 255–461
406
Guihua Gong
of choosing L = n(N + 2), we choose
L ≥ max{n(N + 2), 8δ , 4ε , ε41 }.
Consider ε̃ = 81L < min(ε, ε1 ). Let positive number η̃ <
dist(x, x′ ) < 2η̃, then
ε̃
kf (x) − f (x′ )k < .
3
η
4
satisfy that, if
Let δ̃ > 0 be any number. Then for the pair (η̃, δ̃), there exists an integer L̃
playing the role of L as in Lemma 4.33. We can assume L̃ > L. Let
Λ = 6L̃2 · 2L̃ (dim X + M + 1)3 ,
where M is a positive integer.
Now let Y be a simplicial complex with dim Y ≤ M , and φ : C(X) →
P Mk (C(Y ))P be a unital homomorphism satisfying the following two conditions:
η̃
η
, δ) and sdp( 32
, δ̃);
(a) φ has both sdp( 32
(b) rank(P ) ≥ Λ.
We will construct a decomposition for φ.
η̃
Step 1. By the discussion in 4.34 corresponding to sdp( 32
, δ̃), there is a set
©
ª
Θ(y) = α◦β1 (y)∼L2 , α◦β2 (y)∼L2 , · · · , α◦βL̃−1 (y)∼L2 , α◦βL̃ (y)∼L2 +L1 ,
where
L2 = int
µ
rank(P )
L̃
¶
µ ¶
Λ
≥ int
,
L̃
such that SPφy and Θ(y) can be paired within
η̃
2.
As in 4.34, there are mutually orthogonal projections p0 and P1 =
a homomorphism φ1 : C(X) → P1 Mk (C(Y ))P1 , such that
1
(1) kφ(f ) − p0 φ(f )p0 ⊕ φ1 (f )k ≤ ε̃ ≤ 8L
,
³ ´
L̃
3
(2) rank(p0 ) ≤ L̃ · 2 (dim X + M + 1) ≤ int 6ΛL̃ ,
where φ1 is defined by
φ1 (f )(y) =
L̃
X
PL̃
i=1
pi and
f (α◦βi (y))pi
i=1
with rank(pi ) ≥ L2 − 2L̃ (dim X + M + 1)3 .
Step 2. We will take a part p′ φ1 p′ out from φ1 and add it into p0 φp0 , such
(P )
, which is neither too
that the projection P0 = p0 + p′ has rank about rank
L
large nor too small. (Here we use L not L̃.)
There exists a projection p′ satisfying the following two conditions.
PL̃
(c) p′ = i=1 p′i , with p′i < pi , i = 1, 2, · · · , L̃.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
407
´
³
(P )
(here we use L, not L̃), where L was chosen in
(d) rank(p′ ) = int rank
L
the beginning of this subsection.
We can make the above (d) hold for the following reason. First,
rank(
L̃
X
i=1
pi ) ≥ rank(P ) − int
µ
Λ
6L̃
¶
> int
µ
rank(P )
L
¶
+ L̃ dim(Y ).
PL̃
So ³one can ´
choose non negative integers k1 , k2 , · · · , kL̃ such that
i=1 ki =
(P )
and
that
k
≤
rank(p
)
−
dim(Y
).
Therefore,
by
[Hu],
we can
int rank
i
i
L
choose trivial projections p′i < pi with rank(p′i ) = ki .
Define
P0 = p 0 ⊕ p ′
and
newP1 = P1 ⊖ p′ .
PL̃
Note that p′ is a sub-projection of P1 = i=1 pi . Define newφ1 : C(X) →
newP1 Mk (C(Y ))newP1 by
(newφ1 (f ))(y) =
L̃
X
i=1
f (α◦βi (y))(pi ⊖ p′i ).
newP1 and newφ1 are still denoted by P1 and φ1 , respectively. Evidently, the
following are true.
(1′ )
′
(2 )
kφ(f ) − P0 φ(f )P0 ⊕ φ1 (f )k <
rank(P )
≤ rank(P0 ) ≤ 2 · int
L
µ
1
.
4L
rank(P )
L
¶
.
(Notice that, to get the above decomposition, one only needs the condition
that SPφy and Θ(y) can be paired within η̃ (see the way η is chosen in 4.4
for Theorem 4.1 and the way η̃ is chosen above). On the other hand, SPφy
and Θ(y) can be paired within η̃2 in our case. So if ψ satisfies the condition
that SPψy and SPφy can be paired within η̃2 , then the above decomposition
also holds for ψ, as discussed in 4.41. In particular, for a certain unitary u,
Adu ◦ ψ can have same form of decomposition as φ does— same projection P 0
and even exactly the same part of the above φ1 . This will be used in 4.46 and
Proposition 4.47.)
Step 3. Now, we can decompose φ1 again to obtain a large part of the homomorphism defined by point evaluations, which will be used to absorb the part
of P0 φP0 , in the proof of the uniqueness theorem in [EGL].
For the compact metric space X, and η > 0 (now we use η not η̃), there exists
a finite η-dense subset {x1 , x2 , · · · , xm } such that dist(xi , xj ) ≥ η, if i 6= j.
Documenta Mathematica 7 (2002) 255–461
408
Guihua Gong
(Such set could be chosen to be a maximum set of finite many points which
have mutual distance at least η. Then the η-density of the set follows from the
maximality.)
We will prove the following claim.
Claim: There are mutually orthogonal projections q1 , q2 , · · · , qm < P1 with
rank(qi ) > rank(P0 ) + dim(Y ), such that
°
°
m
m
m
°
°
X
X
X
°
°
f (xi )qi ° < ε
qi ) ⊕
qi )φ1 (f )(P1 −
°φ1 (f ) − (P1 −
°
°
i=1
i=1
i=1
for all f ∈ F .
Proof of the claim:
First, we know that the set (SPφ1 )y is obtained by deleting rank(P0 ) points
(counting multiplicity) from the set Θ(y). Also SPφy and Θ(y) can be paired
within η̃2 . Recall that,
©
ª
Θ(y) = α◦β1 (y)∼L2 , α◦β2 (y)∼L2 , · · · , α◦βL̃−1 (y)∼L2 , α◦βL̃ (y)∼L2 +L1
is the set³ corresponding
to φ and the pair (η̃, δ̃) in 4.33. And recall that
´
rank
(P )
η
L2 = int
, δ). So Θ(y) has the
. From (a), φ has the property sdp( 32
L̃
η
+ η̃2 , δ). But (SPφ1 )y is obtained by deleting
property sdp( 32
µ
µ
¶
¶
rank(P )
δ
rank(P0 ) ≤ 2 · int
≤ rank(P )
L
4
η
+ η̃2 )-ball of any
points from Θ(y). (Note that L1 < 8δ .) Therefore, in the ( 32
point in X, (SPφ1 )y contains at least
δ
3δ
δ · rank(P ) − rank(P ) =
rank(P )
4
4
η
+ η̃2 , 3δ
points (counting multiplicity). That is, φ1 has the property sdp( 32
4 ).
η
η 3δ
Therefore φ1 has the property sdp( 4 , 4 ), since η̃ < 4 .
Set Ui = B η2 (xi ), i = 1, 2, · · · , m. Then Ui , i = 1, 2, · · · , m are mutually
disjoint open sets, since dist(xi , xj ) ≥ η, if i 6= j. By the property sdp( η4 , 3δ
4 )
of φ1 , for any y ∈ Y ,
#(SP(φ1 )y ∩ Ui ) ≥
2
3δ
rank(P ) > rank(P ) + 3 dim Y > rank(P0 ) + 3 dim(Y ).
4
L
The claim follows from the following proposition:
Proposition. Let X be a simplicial complex, and F ⊂ C(X) a finite subset.
Let ε > 0 and η > 0 be such that if dist(x, x′ ) < 2η, then |f (x) − f (x′ )| < 3ε for
any f ∈ F .
Suppose that U1 , U2 , · · · , Um are disjoint open neighborhoods of points
x1 , x2 , · · · , xm ∈ X, respectively, such that Ui ⊂ Bη (xi ) for all 1 ≤ i ≤ m.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
409
Suppose that φ : C(X) → P M• (C(Y ))P is a unital homomorphism, where Y
is a simplicial complex, such that
#(SPφy ∩ Ui ) ≥ ki ,
for 1 ≤ i ≤ m and for all y ∈ Y.
Then there are mutually orthogonal projections q1 , q2 , · · · , qm ∈ P M• (C(Y ))P
with rank(qi ) ≥ ki − dim(Y ) such that
kφ(f ) − p0 φ(f )p0 ⊕
m
X
i=1
f (xi )qi k < ε,
for all f ∈ F,
P
where p0 = P − qi .
This is Proposition 1.5.7 of this paper (see 1.5.4—1.5.6 for the proof). Since the
expert reader may skip §1.5, we point out that the above result was essentially
proved in [EG2, Theorem 2.21].
So we obtain the projections qi with
rank(qi ) ≥ min(#(SP(φ1 )y ∩ Ui )) − dim(Y ) ≥ rank(p0 ) + 2 dim(Y ).
y
Summarizing the above, we obtain the following theorem.
Theorem 4.45. Let F ⊂ C(X) be a finite set, ε > 0, ε1 > 0, and let M be
a positive integer (in the application in [EGL], we will let M = 3). Let the
positive number η < ε41 satisfy that, if dist(x, x′ ) < 2η, then
kf (x) − f (x′ )k <
ε
3
for all f ∈ F.
Let δ > 0 be any positive number. There is an integer L > max{ 8δ , 4ε , ε41 } satisfying the following condition. The rest of the theorem describes this condition.
Suppose that η̃ > 0 satisfies that, if dist(x, x′ ) < 2η̃, then
kf (x) − f (x′ )k <
1
24L
for all f ∈ F.
For any δ̃ > 0, there is a positive integer Λ such that if a unital homomorphism φ : C(X) → P Mk (C(Y ))P (with dim Y ≤ M ) satisfies the following
conditions
η̃
η
, δ) and sdp( 32
, δ̃);
(a) φ has the properties sdp( 32
(b) rank(P ) ≥ Λ,
then there are projections P0 , P1 ∈ P Mk (C(Y ))P (with P0 + P1 = P ) and a
homomorphism φ1 : C(X) → P1 Mk (C(Y ))P1 such that
1
(1) kφ(f ) − P0 φ(f )P0 ⊕ φ1 (f )k < 4L
for all f ∈ F ;
rank
(P )
;
(2) rank(P ) ≥
0
L
(3) There are mutually orthogonal projections q1 , q2 ,· · ·, qm ∈ P1 Mk (C(Y ))P1
and an η-dense finite subset {x1 , x2 , · · · , xm } ⊂ X with the following properties.
Documenta Mathematica 7 (2002) 255–461
410
Guihua Gong
(i) rank(qi ) > rank(PP
1, 2, · · · , m; P
0 ) + 2 dim(Y ), i = P
m
m
m
(ii) kφ1 (f ) − (P1 − i=1 qi )φ1 (f )(P1 − i=1 qi ) ⊕ i=1 f (xi )qi k < ε for all
f ∈ F.
4.46.
Let η̃ and δ̃ be as in 4.44 (or 4.45), and H(η̃, δ̃, X) ⊂ C(X) the
subset defined in 4.27. Suppose that φ : C(X) → P Mk (C(Y ))P satisfies
the conditions (a) and (b) in Theorem 4.45. And suppose that ψ : C(X) →
P Mk (C(Y ))P is another homomorphism satisfying
kAffT φ(h) − AffT ψ(h)k <
δ̃
,
4
for all h ∈ H(η̃, δ̃, X). Similar to 4.41, there is a unitary u ∈ P Mk (C(Y ))P
such that
kAdu◦ψ(f ) − P0 Adu◦ψ(f )P0 ⊕ φ1 k <
1
, ∀f ∈ F
4L
where P0 and φ1 are exactly the same as those for φ in Theorem 4.45. (See the
end of step 2 of 4.44.)
So we have the following proposition.
Proposition 4.47. Let F ⊂ C(X) be a finite set, ε > 0, ε1 > 0, and let M
be a positive integer (in the application in [EGL], we will let M = 3). Let the
positive number η < ε41 satisfy that, if dist(x, x′ ) < 2η, then
kf (x) − f (x′ )k <
ε
3
for all f ∈ F.
Let δ > 0 be any positive number. There is an integer L > max{ 8δ , 4ε , ε41 }
satisfying the following condition. The rest of the proposition describes this
condition.
Suppose that η̃ > 0 satisfies that, if dist(x, x′ ) < 2η̃, then
kf (x) − f (x′ )k <
1
24L
for all f ∈ F.
For any δ̃ > 0, there is a positive integer Λ and a finite set H ⊂ AffT (C(X))(=
C(X)) such that if unital homomorphisms φ, ψ : C(X) → P Mk (C(Y ))P (with
dim Y ≤ M ) satisfy the following conditions:
η̃
η
, δ) and sdp( 32
, δ̃);
(a) φ has the properties sdp( 32
(b) rank(P ) ≥ Λ;
(c) kAffT φ(h) − AffT ψ(h)k < 4δ̃ , ∀h ∈ H,
then there are projections P0 , P1 ∈ P Mk (C(Y ))P (with P0 + P1 = P ), a homomorphism φ1 : C(X) → P1 Mk (C(Y ))P1 factoring through C[0, 1], and a
unitary u ∈ P Mk (C(Y ))P such that
1
(1) kφ(f ) − P0 φ(f )P0 ⊕ φ1 (f )k < 4L
and
1
k(Adu ◦ ψ)(f ) − P0 (Adu ◦ ψ)(f )P0 ⊕ φ1 (f )k < 4L
for all f ∈ F ;
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
411
(P )
(2) rank(P0 ) ≥ rank
;
L
(3) There are mutually orthogonal projections q1 , q2 ,· · ·, qm ∈ P1 Mk (C(Y ))P1
and an η-dense finite subset {x1 , x2 , · · · , xm } ⊂ X with the following properties.
(i) rank(qi ) > rank(P0 ) + 2 dim(Y );
Pm
Pm
Pm
(ii) kφ1 (f ) − (P1 − i=1 qi )φ1 (f )(P1 − i=1 qi ) ⊕ i=1 f (xi )qi k < ε for all
f ∈ F.
In order to be consistent in notation with the application in [EGL], let us
rewrite the above proposition in the following form.
Proposition 4.47’. For any finite set F ⊂ C(X), ε > 0, ε1 > 0, there is a
number η > 0 with the property described below.
For any δ > 0, there are an integer K > 4ε and a a number η̃ > 0 satisfying
the following condition.
For any δ̃ > 0, there is a positive integer L and a finite set H ⊂ AffT (C(X))(H
can be chosen to be H(η̃, δ̃, X) in 4.27) such that if two unital homomorphisms
φ, ψ : C(X) → P Mk (C(Y ))P (with dim Y ≤ 3) satisfy the following conditions:
η̃
η
, δ) and sdp( 32
, δ̃);
(a) φ has the properties sdp( 32
(b) rank(P ) ≥ L;
(c) kAffT φ(h) − AffT ψ(h)k < 4δ̃ , ∀h ∈ H,
then there are projections P0 , P1 ∈ P Mk (C(Y ))P (with P0 + P1 = P ), a homomorphism φ1 : C(X) → P1 Mk (C(Y ))P1 factoring through C[0, 1], and a
unitary u ∈ P Mk (C(Y ))P such that
1
(1) kφ(f ) − P0 φ(f )P0 ⊕ φ1 (f )k < 4K
and
1
for all f ∈ F ;
k(Adu ◦ ψ)(f ) − P0 (Adu ◦ ψ)(f )P0 ⊕ φ1 (f )k < 4K
rank
(P )
;
(2) rank(P ) ≥
0
K
(3) There are mutually orthogonal projections q1 , q2 ,· · ·, qm ∈ P1 Mk (C(Y ))P1
and an ε41 -dense finite subset {x1 , x2 , · · · , xm } ⊂ X with the following properties.
(i) rank(qi ) > rank(P0 ) + 2 dim(Y );
Pm
Pm
Pm
(ii) kφ1 (f ) − (P1 − i=1 qi )φ1 (f )(P1 − i=1 qi ) ⊕ i=1 f (xi )qi k < ε for all
f ∈ F.
(Notice that in the above statement, we change the notation L and Λ to K and
L respectively. Also, in condition (3), we change η-density to ε41 -density.)
4.48. Proposition 4.47’ will be used in the proof of the Uniqueness Theorem
in [EGL]. Namely, we will prove that, under certain conditions about KK(φ)
and KK(ψ) and the determinants of φ(z) and ψ(z) (see (4) of Theorem 2.4 of
[NT] ), where z ∈ C(S 1 ) is the standard generator,
P0 φ(f )P0 ⊕
m
X
f (xi )qi ,
i=1
Documenta Mathematica 7 (2002) 255–461
f ∈F
412
Guihua Gong
is approximately unitarily equivalent to
P0 Adu◦ψ(f )P0 ⊕
m
X
f ∈ F.
f (xi )qi ,
i=1
Therefore, {φ(f ), f ∈ F } is approximately unitarily equivalent to {ψ(f ), f ∈
F }. In [EGL], we need both of the following conditions:
rank(P0 ) ≥
rank(P )
L
and
[qi ] > [P0 ] in K0 (C(Y )).
In comparison with Theorem 2.4 of [NT], in the Uniqueness Theorem in [EGL],
we also have a condition similar to (4) of Theorem 2.4 of [NT]. But this condition will be useful only when it is combined with the condition (2) above (see
[EGL] for details).
5
Almost Multiplicative Maps
In this section, we study almost multiplicative maps
φ ∈ Map (Ml (C(X)), Ml1 (C(Y ))) ,
where X = TII,k , TIII,k , or S 2 , and Y is a simplicial complex of dimension at
most M with M a fixed number. In this section, all the simplicial complexes
are assumed to have dimension at most M .
Suppose that B1 , B2 , · · · , Bn , · · · are unital Q
C ∗ -algebras. Let B =
+∞
n=1 Bn . Then the multiplier algebra M (B) of B is
n=1 Bn . The Six Term
Exact Sequence associated to
5.1.
L+∞
0 −→ B −→ M (B) −→ M (B)/B −→ 0
breaks into two exact sequences
0 −→ K0 (B) −→ K0 (M (B)) −→ K0 (M (B)/B) −→ 0
and
0 −→ K1 (B) −→ K1 (M (B)) −→ K1 (M (B))/B −→ 0.
since each projection (or unitary) in Mn (M (B)/B) can be lifted to a projection
(or a unitary) in Mn (M (B)).
Furthermore,
K0 (B) =
+∞
M
K0 (Bn )
and
K1 (B) =
+∞
M
K1 (Bn ).
n=1
n=1
But in general, it is not true that
K0 (M (B)) =
+∞
Y
n=1
K0 (Bn )
or
K1 (M (B)) =
+∞
Y
n=1
Documenta Mathematica 7 (2002) 255–461
K1 (Bn ).
Simple Inductive Limit C ∗ -Algebras, I
413
Q+∞
In fact, K0 (M (B)) is a subgroup of n=1 K0 (Bn ). But K1 (M (B)) is more
complicated. In the first part of this section, we will calculate the K-theory of
M (B) (and of M (B)/B) for the case
Bn = Mkn (C(Xn )),
where Xn are simplicial complexes of dimension at most M . For convenience,
we always suppose that the spaces Xn are connected.
| . Let
5.2. Consider S 1 = {z; |z| = 1} ⊂ C
F : (S 1 \{−1}) × [0, 1] −→ S 1 \{−1}
be defined by
F (eiθ , t) = eitθ ,
−π < θ < π.
Then |t − t′ | < ε implies
|F (x, t) − F (x, t′ )| < πε.
This fact implies the following. If u and v are unitaries such that ku − vk < 1,
then there is a path of unitaries ut with u0 = u, u1 = v such that |t − t′ | < ε
implies kut − ut′ k < πε.
Let SU (n)(⊂ U (n)) denote the collection of n × n unitaries with determinant
1. Let SUn (X) denote the collection of continuous functions from X to SU (n).
Note SUn (X) ⊂ Un (X) ⊂ Mn (C(X)).
From the proof of Theorem 3.3 (and Lemma 3.1) of [Phi2] (in particular (∗ ∗ ∗)
in Step 4 of 3.3 of [Phi2]), one can prove the following useful fact.
Lemma 5.3. ([Phi2]) For each positive integer M , there is an M ′ > 0 satisfying the following condition. For any connected finite CW-complex X of
dimension at most M , and u, v ∈ SUn (X), if u and v can be connected to each
other in Un (X), then there is a path ut ∈ SUn (X) such that
1. u0 = u, u1 = v and
2. |t − t′ | < ε implies kut − ut′ k < M ′ · ε.
(Note that M ′ does not depend on n, the size of the unitaries.)
L+∞
5.4. Let Bn = Mkn (C(Xn )), dim(Xn ) ≤ M . Let B = n=1 Bn . Then we
can describe K0 (M (B)) as below. Let (K0 (Bn ), K0 (Bn )+ , 1Bn ) be the scaled
ordered
Q+∞ K-group of Bn (see 1.2 of [EG2]). Let Πb K0 (Bn ) be the subgroup of
n=1 K0 (Bn ) consisting of elements
(x1 , x2 , · · · , xn , · · ·) ∈
+∞
Y
K0 (Bn )
n=1
with the property that there is a positive integer L such that
−L[1Bn ] < xn < L[1Bn ] ∈ K0 (Bn )
Documenta Mathematica 7 (2002) 255–461
414
Guihua Gong
for all n.
´
Bn = Πb K0 (Bn ).
´
³Q
+∞
is of the form [p] − [q], where p, q ∈
B
Proof: Any element in K0
n=1 n
´
³Q
+∞
ML
n=1 Bn are projections. Let
Lemma 5.5. K0
³Q
+∞
n=1
p = (p1 , p2 , · · · , pn , · · ·), q = (q1 , q2 , · · · , qn , · · ·) ∈ ML
Then [p] − [q] ∈ K0
³Q
+∞
n=1
´
Bn corresponds to the element
à +∞
Y
Bn
n=1
!
.
([p1 ] − [q1 ], [p2 ] − [q2 ], · · · , [pn ] − [qn ], · · ·) ∈ Πb K0 (Bn ).
We will prove that this correspondence is bijective.
Surjectivity: Let
([p1 ] − [q1 ], [p2 ] − [q2 ], · · · , [pn ] − [qn ], · · ·) ∈ Πb K0 (Bn ).
Then there is an L > M such that
−L[1Bn ] < [pn ] − [qn ] < L[1Bn ], ∀n.
Therefore,
−L · kn ≤ rank(pn ) − rank(qn ) < L · kn , ∀n.
It is well known that (see [Hu]) any vector bundle of dimension M + T over
an M dimensional space has a T dimensional trivial sub-bundle. Thus one can
replace pn by p′n , qn by qn′ , with properties
[p′n ] < 2L[1Bn ], [qn′ ] < 2L[1Bn ]
[p′n ]
−
[qn′ ]
= [pn ] − [qn ]
and
in K0 (Bn ).
([p′1 ] − [q1′ ], [p′2 ] − [q2′ ], · · ·) is in the image of the correspondence, since every
element [p′n ] < 2L[1Bn ] can be realized by a projection in M4L (Bn ) (recall that
L > M ).
Injectivity.
Let ³p = (p1 ,´p2 , · · · , pn , · · ·) and q = (q1 , q2 , · · · , qn , · · ·) be
Q+∞
Suppose that for each n, [pn ] = [qn ] ∈
projections in ML
n=1 Bn .
K0 (B
have to prove that [(p1 , p2 , · · · , pn , · · ·)] = [(q1 , q2 , · · · , qn , · · ·)] ∈
n ). We ´
³Q
+∞
K0
n=1 Bn .
Without loss of generality, assume that L > M . Let 1n ∈ ML (Bn ) be the unit.
By [Hu], for each n, the projection pn ⊕ 1n is unitary equivalent to qn ⊕ 1n .
That is, there is a unitary un ∈ M2L (Bn ) such thatQqn ⊕ 1n = un (pn ⊕ 1n )u∗n .
+∞
Hence the unitary u = (u1 , u2 , · · · , un , · · ·) ∈ M2L ( n=1 Bn ) satisfies q ⊕ 1 =
u(p ⊕ 1)u∗ . It follows that [q] = [p].
⊔
⊓
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
415
5.6. Let X be a finite CW complex. Then K1 (C(X)) = K 1 (X) is defined to
be the collection of homotopy equivalence classes of continuous maps from X
to U (∞), denoted by [X, U (∞)], (or from X to U (n), denoted by [X, U (n)],
for n large enough). Consider the fibration
b
SU (n) −→ U (n) −→ S 1 ,
| is the unit circle, b is defined by sending a unitary to its dewhere S 1 ⊂ C
terminant, and SU (n) is the special unitary group consisting the unitaries of
b−1
determinant 1. The fibration has a splitting S 1 −→ U (n), defined by
z
1
b−1
S 1 ∋ z 7−→
∈ U (n).
..
.
1
One can identify U (n) with SU (n) × S 1 by U (n) ∋ u 7→ ((b−1 ◦ b(u))∗ u, b(u)) ∈
SU (n) × S 1 .
Therefore, [X, U (n)] = [X, SU (n)] ⊕ [X, S 1 ] as a group. We use notation
SK1 (C(X)) or SK 1 (X) to denote [X, SU (n)], n large enough, and π 1 (X) to
denote [X, S 1 ]. Then
K1 (C(X)) = SK1 (C(X)) ⊕ π 1 (X).
(The splitting is not a natural splitting.)
5.7. Let {Xn } be a sequence of connected finite CW complexes of dimension
at most M . Let Bn = Mkn (C(Xn )). Define a map
!
à +∞
+∞
Y
Y
K1 Bn
by
Bn −→
τ : K1
n=1
n=1
τ [(u1 , u2 , · · · , un , · · ·)] = ([u1 ], [u2 ], · · · , [un ], · · ·),
´
³Q
+∞
. If L ≥ M , then any
B
where (u1 , u2 , · · · , un , · · ·) is a unitary in ML
n
n=1
element in K1 (Bn ) can be realized by a unitary in ML (Bn ). Based on this fact,
we know that τ is surjective. We will prove that
!
à +∞
+∞
Y
Y
K1 Bn −→ 0
Bn −→
0 −→ Kerτ −→ K1
n=1
n=1
is a splitting exact sequence. A splitting
τ̃ :
+∞
Y
n=1
K1 Bn −→ K1
à +∞
Y
n=1
Bn
!
Documenta Mathematica 7 (2002) 255–461
416
Guihua Gong
will be defined such that τ ◦ τ̃ = id on
Q+∞
n=1
K1 Bn . By 5.6,
K1 Bn = SK1 Bn ⊕ π 1 (Xn ).
Q+∞
Q+∞
Hence we define τ̃ on n=1 SK1 Bn and n=1 π 1 (Xn ) separately.
Q+∞
If x ∈ n=1 SK1 Bn is represented by a sequence of unitaries
u1 ∈ ML (B1 ), u2 ∈ ML (B2 ), · · · , un ∈ ML (Bn ), · · · ,
each with determinant 1, then define
τ̃ (x) = [(u1 , u2 , · · · , un , · · ·)] ∈ K1
à +∞
Y
n=1
Bn
!
.
To see that τ̃ is well defined, let v1 , v2 , · · · , vn , · · · be another sequence with
determinant 1 and
[un ] = [vn ]
in K1 (Bn ).
Without loss of generality, we assume that L > M . By Lemma 5.3, for each
n, there is a unitary path un such that un (0) = un , un (1) = vn , and kun (t) −
un (t′ )k < M ′ ·|t−t′ |, ∀t ∈ [0, 1], where M ′ is a constant which does not depend
on n. Obviously,
(u1 (t), u2 (t), · · · , un (t), · · ·) ∈ (ML (
+∞
Y
n=1
Bn )) ⊗ C([0, 1]).
Hence
[(u1 , u2 , · · · , un , · · ·)] = [(v1 , v2 , · · · , vn , · · ·)].
That is, the above map is well defined.
(Warning: It is not enough to prove that each un can be connected to vn , since
a sequence of paths,
Q+∞ each connecting un and vn (n = 1, 2, · · ·), only defines an
element in ML ( n=1 (Bn ⊗ C[0, 1])), but
!
à +∞
+∞
Y
Y
(Bn ⊗ C[0, 1]). )
Bn ⊗ C[0, 1] ⊂
6=
n=1
n=1
The following claim is a well known folklore result in topology. Since we can
not find a precise reference, we present a proof here.
Claim: For any connected simplicial complex X, the cohomotopy group π 1 (X)
is a finitely generated free abelian group.
Proof of the claim. Let X (1) be the 1-skeleton of X. Then X (1) is homotopy
equivalent to a finite wedge of S 1 . Evidently, π 1 (X (1) ) is a finitely generated
free abelian group. (In comparison with the above cohomotopy group, we point
out that the fundamental group π1 (X (1) ) of a finite wedge X (1) of S 1 is a free
group (not a free abelian group).)
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
417
On the other hand, we can prove that
i∗ : π 1 (X) → π 1 (X (1) ),
induced by the inclusion i : X (1) → X, is an injective map as below. Once this
is done, the claim follows from the result in group theory that any subgroup of
a free abelian group is still a free abelian group.
Let us prove the injectivity of i∗ . Suppose that f, g : X → S 1 are two maps
satisfying that
i∗ ([f ]) = i∗ ([g]),
where [f ], [g] ∈ π 1 (X) are elements represented by f and g, respectively. Then
f |X (1) is homotopic to g|X (1) . Let F : X (1) × [0, 1] → S 1 be a homotopy path
connecting f |X (1) and g|X (1) . That is
F |X (1) ×{0} = f |X (1) and
F |X (1) ×{1} = g|X (1) .
We are going to extend the homotopy F to a homotopy on the entire space
X × [0, 1]. The construction is done by induction. Suppose that F has been
extended to a homotopy (let us still denote it by F ) F : X (n) × [0, 1] →
S 1 between f |X (n) and g|X (n) on the n-skeleton (where n ≥ 1) of X. I.e.,
F |X (n) ×{0} = f |X (n) and F |X (n) ×{1} = g|X (n) . We need to prove that it can be
extended to a homotopy on the (n + 1)-skeleton. Let ∆ be any (n + 1)-simplex.
Then ∂∆ ⊂ X (n) . Let G : ∂∆ × [0, 1] ∪ ∆ × {0} ∪ ∆ × {1} → S 1 be defined by
F (x)
f (x)
G(x) =
g(x)
if x ∈ ∂∆ × [0, 1]
if x ∈ ∆ × {0}
if x ∈ ∆ × {1} .
Then G(x) is a continuous map from ∂(∆ × [0, 1]) to S 1 . Since πn+1 (S 1 ) = 0
and ∂(∆ × [0, 1]) = S n+1 , G can be extended to a map G : ∆ × [0, 1] → S 1 .
Define F on each simplex ∆ to be this G. Then F is the desired extension.
This ends the proof of the claim.
Q+∞
Let us go back to the construction of τ̃ on n=1 π 1 (Xn ). Let x01 ∈ X1 , x02 ∈
X2 , · · · , x0n ∈ Xn , · · · , be chosen as the base points of the spaces. Let
θn,1 , θn,2 , · · · , θn,tn : Xn −→ S 1
be the functions representing the generators
[θn,1 ], [θn,2 ], · · · , [θn,tn ] ∈ π 1 (Xn ).
Suppose that
| ,
θn,j (x0n ) = 1 ∈ S 1 ⊂ C
j = 1, 2, · · · , tn .
Documenta Mathematica 7 (2002) 255–461
418
Guihua Gong
Q+∞
For any element (x1 , x2 , · · · , xn , · · ·) ∈ n=1 π 1 (Xn ), define
un ∈ Bn be defined by
θn,1 (y)m1 · θn,2 (y)m2 · · · θn,tn (y)mtn
1
un (y) =
..
.
1
τ̃ (x) as below. Let
for each y ∈ Xn , where m1 , m2 , · · · , mtn are integers with
(∗)
| )
∈ Mkn (C
kn ×kn
xn = m1 [θn,1 ] + m2 [θn,2 ] + · · · + mtn [θn,tn ] ∈ π 1 (Xn ).
Define
τ̃ (x) = [(u1 , u2 , · · · , un , · · ·)] ∈ K1
à +∞
Y
Bn
n=1
!
.
Since each π 1 (Xn ) is a freeQabelian group, the expression (∗) for xn is unique. It
+∞
is easy to check that τ̃ on n=1 π 1 (Xn ) is a well defined group homomorphism.
It is straight forward to check that
τ ◦ τ̃ = id :
And that
τ ◦ τ̃ = id :
That is,
τ ◦ τ̃ = id :
The splitting τ̃ :
Q+∞
+∞
Y
n=1
+∞
Y
n=1
π (Xn ) −→
SK1 (Bn ) −→
+∞
Y
n=1
K1 (Bn ) −→
n=1 K1 (Bn ) −→ K1
0 −→ Ker(τ ) −→ K1
+∞
Y
1
³Q
à +∞
Y
+∞
n=1
Bn
n=1
!
π 1 (Xn )
n=1
+∞
Y
SK1 (Bn ).
+∞
Y
K1 (Bn ).
n=1
n=1
´
Bn of the exact sequence
τ
−→
+∞
Y
n=1
K1 Bn −→ 0
gives an isomorphism
K1
à +∞
Y
n=1
Bn
!
=
+∞
Y
n=1
K1 Bn ⊕ Ker(τ ).
5.8. In order to identify Ker(τ ), suppose that
u = [(u1 , u2 , · · · , un , · · ·)] ∈ K1
à +∞
Y
n=1
Bn
!
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
satisfies that
τ (u) = 0 ∈
+∞
Y
419
K1 Bn .
n=1
| ) can be connected to 1 ∈ M• (C
| ), by
Note that any unitary matrix v ∈ M• (C
a path v(t) satisfying that, if |t − t′ | < ε, then
kv(t) − v(t′ )k < 2πε.
Based on this fact, we have
[(u1 , u2 , · · ·)] =
[(u∗1 (x01 )u1 , u∗2 (x02 )u2 , · · ·)]
∈ K1
à +∞
Y
n=1
Bn
!
.
Therefore, without loss of generality, we assume that
un (x0n ) = 1 ∈ ML (Bn ),
where x0n ∈ Xn are the base points.
Since τ (u) = 0, if we assume L ≥ M , then each un can be connected to
1 ∈ ML (Bn ). This implies that the map
determinant(un ) : Xn −→ S 1
is homotopy trivial. Therefore, this map can be lifted to a unique map
det(un ) : Xn −→ IR
such that det(un )(x0n ) = 0 ∈ IR and
exp(2πi det(un )) = determinant(un ).
`
`
Let Xn be the `
disjoint union of Xn and Map ( Xn , IR)0 the set `
of all continuous maps f : Xn → IR with f (x0n ) = 0 for all x0n . Let Mapb ( Xn , IR)0
be the set of those maps with bounded
¢
¡` images.
Xn ,IR
Map +∞
n=1
¡
¢0 by
`
Define a map d : Ker(τ ) →
Mapb +∞
Xn ,IR
n=1
0
d(u) = [(
det(un )
det(u1 ) det(u2 )
,
,···,
, · · ·)].
k1
k2
kn
We will prove that d is a well defined isomorphism.
Suppose that u can be represented by another unitary
!
à +∞
Y
Bn
(v1 , v2 , · · · , vn , · · ·) ∈ ML
n=1
Documenta Mathematica 7 (2002) 255–461
420
Guihua Gong
with v(x0n ) = 1 ∈ ML (Bn ). Then for the unit of a certain matrix algebra over
Q
+∞
n=1 Bn
!
à +∞
Y
1L1 ∈ ML1
Bn ,
n=1
we have that, the element
(u1 ⊕ 1, u2 ⊕ 1, · · · , un ⊕ 1, · · ·) ∈ ML+L1
à +∞
Y
Bn
n=1
!
can be connected to the element
(v1 ⊕ 1, v2 ⊕ 1, · · · , vn ⊕ 1, · · ·) ∈ ML+L1
à +∞
Y
Bn
n=1
!
by a unitary path
(u1 (t), u2 (t), · · · , un (t), · · ·) ∈
Ã
ML+L1
à +∞
Y
Bn
n=1
!!
⊗ C[0, 1].
We need to prove that
µ
¶
det(un ) − det(vn )
det(u1 ) − det(v1 ) det(u2 ) − det(v2 )
,
,···,
,···
k1
k2
kn
has a uniformly bounded image in IR. This follows from the following fact. If
| ) satisfying |w1 − w2 | < ε < 1 , then
two unitaries w1 , w2 ∈ M(L+L1 )kn (C
4
|determinant(w1∗ w2 ) − 1| < π(L + L1 )kn ε.
Now, we have to prove that d is an isomorphism.
Obviously, d is surjective. In fact, for any function
à +∞ !
a
(f1 , f2 , · · · , fn , · · ·) ∈ Map
IR ,
n=1
let u ∈ K1
³Q
+∞
n=1
´
0
Bn be the element represented by
(exp(2πif1 ), exp(2πif2 ), · · · , exp(2πifn ), · · ·) ∈
+∞
Y
n=1
Bn =
+∞
Y
Mkn (C(Xn )).
n=1
Then d(u) = [(f1 , f2 , · · · , fn , · · ·)].
Finally, we have to prove that d is injective. Suppose that u ∈ Ker(τ ) is
represented by
!
à +∞
Y
Bn
(u1 , u2 , · · · , un , · · ·) ∈ ML
n=1
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
satisfying
µ
Let fn =
det(un )
det(u1 ) det(u2 )
,
,···,
,···
k1
k2
kn
det(un )
kn
¶
∈ Mapb
³a
421
Xn , IR
´
0
.
: Xn → IR and
vn = exp
2πifn
∈ MLkn (C(Xn )).
L
Then vn∗ un ∈ SULkn (Xn ), i.e., it has determinant 1 every where.
(f1 , f2 , · · · , fn , · · ·) is of uniformly bounded image, we know that
(u1 , u2 , · · · , un , · · ·)
Since
(v1∗ u1 , v2∗ u2 , · · · vn∗ un , · · ·)
and
can be connected by a continuous path in
à +∞
!!
Ã
Y
⊗ C[0, 1].
ML
Bn
n=1
Therefore, u = [(v1∗ u1 , v2∗ u2 , · · · vn∗ un , · · ·)]. The latter is zero by Lemma 5.3
and the fact that u ∈ Ker(τ ).
Summarizing the above, we obtain
¢
¡`
´ Q
³Q
Xn ,IR
Map +∞
+∞
+∞
n=1
¡`+∞
¢0
Lemma 5.9. K1
n=1 K1 Bn ⊕ Map
n=1 Bn =
Xn ,IR
b
n=1
0
=
+∞
Y
n=1
SK1 Bn ⊕
+∞
Y
n=1
π 1 (Xn ) ⊕
Map
Mapb
³`
+∞
n=1
³`
+∞
n=1
Xn , IR
´
Xn , IR
´0 .
0
Corollary 5.10.
K1
à +∞
Y
n=1
Bn
+∞
.M
n=1
Bn
!
=
à +∞
Y
n=1
K1 Bn
+∞
.M
n=1
K1 Bn
!
Map
⊕
Mapb
³`
+∞
n=1
³`
+∞
n=1
Xn , IR
´
Xn , IR
´0 .
0
5.11. From [Sch], for any C ∗ -algebra A in the bootstrap class and any C ∗ algebra B (not necessarily separable), there is a splitting short exact sequence
0 −→ K∗ (A) ⊗ K∗ (B) −→ K∗ (A ⊗ B) −→ T or(K∗ (A), K∗ (B)) −→ 0.
Let A = C0 (Wk ), where Wk = TII,k as in the introduction. Wk is used for
TII,k only when involving mod k K-theory K∗ (B, ZZ/k). From the definition
K∗ (B, ZZ/k) := K∗ (A ⊗ B),
Documenta Mathematica 7 (2002) 255–461
422
Guihua Gong
one has
0 −→ K0 (B) ⊗ ZZ/k −→ K0 (B, ZZ/k) −→ T or(ZZ/k, K1 (B)) −→ 0
and
0 −→ K1 (B) ⊗ ZZ/k −→ K1 (B, ZZ/k) −→ T or(ZZ/k, K0 (B)) −→ 0.
Since G ⊗ ZZ/k can be identified with the cokernel of
×k
G −→ G,
and Tor(ZZ/k, G) can be identified with the kernel of
×k
G −→ G,
one has the following well known exact sequences
×k
×k
×k
×k
K0 (B) −→ K0 (B) −→ K0 (B, ZZ/k) −→ K1 (B) −→ K1 (B)
and
K1 (B) −→ K1 (B) −→ K1 (B, ZZ/k) −→ K0 (B) −→ K0 (B).
5.12. Let {Xn }+∞
n=1 , Bn = Mkn (C(Xn )) be as above, and B =
Q+∞
M (B) = n=1 Bn , Q(B) = M (B)/B.
From Lemma 5.5 and Corollary 5.10, we have
L+∞
n=1
Bn ,
K0 (Q(B)) = Πb K0 (Bn )/ ⊕ K0 (Bn )
and
³
´
`+∞
(Q
)
+∞
Map
X
,
IR
M
n
n=1
K1 (Bn )
³`
´0 .
K1 (Q(B)) = Ln=1
+∞
+∞
Map
X , IR
n=1 K1 (Bn )
b
n=1
n
0
It is easy to see that the map
³`
´
³`
´
+∞
+∞
Map
X
,
IR
Map
X
,
IR
n=1 n
n=1 n
×k
³`
³`
´0
´0
−→
+∞
+∞
Map
Map
b
b
n=1 Xn , IR
n=1 Xn , IR
0
0
is an isomorphism.
Any torsion element xn ∈ K0 (Bn ) can be realized as a formal difference of two
projections p, q ∈ M∞ (Bn ) of the same rank. (The rank of a projection makes
sense since Xn are connected. Also for any element x ∈ K0 (Bn ) represented
by [p] − [q], we define rank(x) = rank(p) − rank(q), which is always a (possibly
negative) integer.) By [Hu], if a projection p ∈ M• (C(Xn )) has rank larger
than dim(Xn ) + r, then p has a trivial sub projection of rank r. Therefore,
any torsion element xn ∈ K0 (Bn ) can be realized as a formal difference of two
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
423
projections p, q ∈ ML (Bn ) if L > dim(Xn ). Based on this fact, one can directly
compute that
!
à +∞
+∞
³
´
Y
Y
×k
×k
Kernel Πb K0 (Bn ) −→ Πb K0 (Bn ) = Kernel
K0 (Bn ) .
K0 (Bn ) −→
n=1
n=1
Q+∞
Fixed a positive integer k. let x = (x1 , x2 , · · · , xn , · · ·) ∈ n=1 K0 (Bn ). For
each element xn ∈ K0 (Bn ), one can write rank(xn ) = k · M · ln + rn , where
ln is a (possibly negative) integer, M is the maximum of {dim(Xn )}n , and
0 < k · M ≤ rn < 2k · M . Let pn be the trivial rank one projection in Bn . Then
xn can be written as k · M · ln [pn ] + [qn ], where qn is a projection
of rank rn .
Q+∞
Therefore, x can can be written as x = x′ + x′′ , where x′ ∈ k( n=1 K0 (Bn ))
and x′′ ∈ Πb K0 (Bn ). As a consequence, one can compute that
³
´
×k
Cokernel Πb K0 (Bn ) −→ Πb K0 (Bn )
à +∞
!
+∞
Y
Y
×k
= Cokernel
K0 (Bn ) .
K0 (Bn ) −→
n=1
n=1
Combined with 5.11, yields
K0 (Q(B), ZZ/k) =
+∞
Y
K0 (Bn , ZZ/k)
+∞
Y
K1 (Bn , ZZ/k)
n=1
and
K1 (Q(B), ZZ/k) =
+∞
.M
K0 (Bn , ZZ/k),
+∞
.M
K1 (Bn , ZZ/k).
n=1
n=1
n=1
5.13. Following [DG], denote
K(A) = K∗ (A) ⊕
+∞
M
K∗ (A, ZZ/n).
n=2
For any finite CW complex X and two KK-elements α, β ∈ KK(C(X), A),
from [DL] (also see [DG]), we know that α = β if and only if
α∗ = β∗ : K(C(X)) −→ K(A).
We will discuss the special cases of X = {pt}, [0, 1], TII,k , TIII,k and S 2 , where
TII,k , TIII,k are defined in the Introduction. (See §4 of [EG2] for details.) (The
case X = {pt} or [0, 1] is similar to the case X = S 2 , so we will not discuss the
spaces {pt} and [0, 1] separately.)
From [DL], there is an isomorphism
KK(C(X), B) −→ HomΛ (K(C(X)), K(B)),
Documenta Mathematica 7 (2002) 255–461
424
Guihua Gong
where HomΛ (K(C(X)), K(B)) is the set of systems of group homomorphisms
which is compatible with all the Bockstein Operations (see [DL] for details).
For any fixed finite CW complex X, an element α ∈ KK(C(X), B) is determined by the system of maps
αn∗ : K∗ (C(X), ZZ/n) −→ K∗ (B, ZZ/n), n = 0, 2, 3, · · ·
which are induced by α. In fact α would be determined by a few maps from
the above list—all the other maps in the system {αn∗ }+∞
n=0 : K(C(X)) → K(B)
would be completely determined by these few maps via the Bockstein Operations. We will choose those few maps for the cases X = {pt}, [0, 1], S 2 , TII,k ,
or TIII,k .
1. X = S 2 . Then
K(C(S 2 )) −→ K(B)
is completely determined by
K0 (C(S 2 )) −→ K0 (B)
via the Bockstein Operation
K0 (C(S 2 )) −→ K0 (C(S 2 ), ZZ/k)
↓
↓
K0 (B)
−→
K0 (B, ZZ/k)
since the top horizontal map is surjective. (Note that K1 (C(S 2 )) and
K1 (C(S 2 ), ZZ/k) are trivial groups.) Therefore,
KK(C(S 2 ), B) ∼
= Hom(K0 (C(S 2 )), K0 (B)).
(This is also a well known consequence of the Universal Coefficient Theorem.)
(The case X = {pt} or [0, 1] is similar to the above case.)
| and let C0 (TII,k ) be the ideal of C(TII,k )
2. X = TII,k . Let rC(TII,k ) ∼
=C
consisting of the continuous functions vanishing at the base point. (See 1.6 of
[EG2] and 1.1.7 for the notations.) Consider the splitting exact sequence
0 −→ K0 (C0 (TII,k )) −→ K0 (C(TII,k )) −→ K0 (rC(TII,k )) −→ 0.
Each KK-element α ∈ KK(C(TII,k ), B) induces two group homomorphisms
α00 : K0 (rC(TII,k )) (= ZZ) −→ K0 (B)
and
αk1 : K1 (C(TII,k ), ZZ/k) (= ZZ/k) −→ K1 (B, ZZ/k).
This induces a map
KK(C(TII,k ),B) −→Hom(K0 (rC(TII,k )),K0 (B))
= Hom(ZZ,K0 (B))
L
L
Hom(K1 (C(TII,k ),ZZ/k),K1 (B,ZZ/k))
Hom(ZZ/k,K1 (B,ZZ/k)).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
425
It can be verified that any two homomorphisms
α00 : K0 (rC(TII,k )) (= ZZ) −→ K0 (B)
and
αk1 : K1 (C(TII,k ), ZZ/k) (= ZZ/k) −→ K1 (B, ZZ/k)
induces a unique system of homomorphisms in HomΛ (K(C(TII,k )), K(B)).
Therefore, the above map is an isomorphism.
Another way to see it, is as follows. Note that
K1 (C(TII,k ), ZZ/k) = K0 (C0 (TII,k )) ⊂ K0 (C(TII,k )).
Considering
×k
×k
K1 (B) −→ K1 (B) −→ K1 (B, ZZ/k) −→ K0 (B) −→ K0 (B),
we obtain
Hom(K1 (C(TII,k ), ZZ/k), K1 (B, ZZ/k))
∼
= Hom(K0 (C0 (TII,k )), K0 (B)) ⊕ Ext(K0 (C0 (TII,k )), K1 (B)).
Then from the Universal Coefficient Theorem,
KK(C(TII,k ), B)
∼ Hom(K0 (C(TII,k )), K0 (B)) ⊕ Ext(K0 (C(TII,k )), K1 (B))
=
∼
= Hom(K0 (rC(TII,k )), K0 (B)) ⊕ Hom(K0 (C0 (TII,k )), K0 (B))
⊕Ext(K0 (C0 (TII,k )), K1 (B)).
(Note that K1 (C(TII,k )) = 0.) Hence one can see again, the map mentioned
above is an isomorphism.
| and let C0 (TIII,k ) be the ideal
3. X = TIII,k . Also, let rC(TIII,k ) = C
consisting of functions vanishing at the base point. Notice that
K0 (C(TIII,k )) = ZZ
and
K0 (C0 (TIII,k ), ZZ/k) = ZZ/k.
By the splitting exact sequence
0 → K0 (C0 (TIII,k ), ZZ/k) → K0 (C(TIII,k ), ZZ/k) → K0 (rC(TIII,k ), ZZ/k) → 0
we know that each α ∈ KK(C(TIII,k ), B) induces an element
αk0 : K0 (C0 (TIII,k ), ZZ/k) −→ K0 (B, ZZ/k).
It can be proved that
KK(C(TIII,k ), B)
M
∼
Hom(K0 (C0 (TIII,k ), ZZ/k), K0 (B, ZZ/k))
= Hom(K0 (C(TIII,k )), K0 (B))
M
= Hom(ZZ, K0 (B))
Hom(ZZ/k, K0 (B, ZZ/k)),
Documenta Mathematica 7 (2002) 255–461
426
Guihua Gong
as what we did for TII,k .
(Notice that the map K1 (C(TIII,k )) → K1 (B) is completely determined by the
map K0 (C0 (TIII,k ), ZZ/k) → K0 (B, ZZ/k).)
Summarizing the above, we have the following.
For any two elements α, β ∈ KK(C(X), B), α = β if and only if
(1) α00 = β00 : K0 (C(X)) −→ K0 (B), when X = S 2 ;
(2) α00 = β00 : K0 (rC(X)) −→ K0 (B) and αk1 = βk1 : K1 (C(X), ZZ/k) −→
K1 (B, ZZ/k), when X = TII,k ;
(3) α00 = β00 : K0 (C(X)) −→ K0 (B) and αk0 = βk0 : K0 (C0 (X), ZZ/k) −→
K0 (B, ZZ/k), when X = TIII,k .
Therefore, we have the following lemma.
Lemma 5.14. Let A = P Ml (C(X))P , and X one of {pt}, [0, 1], TII,k , TIII,k ,
or S 2 . Let α, β ∈ KK(A, B), where B is a C ∗ -algebra. Then α = β if and
only if the following hold:
1. When X = {pt}, [0, 1] or S 2 ,
α∗ = β∗ : K0 (A) −→ K0 (B);
2. When X = TII,k ,
α∗ = β∗ : K0 (A) −→ K0 (B)
and
α∗ = β∗ : K1 (A, ZZ/k) −→ K1 (B, ZZ/k);
3. When X = TIII,k ,
α∗ = β∗ : K0 (A) −→ K0 (B)
and
α∗ = β∗ : K0 (A, ZZ/k) −→ K0 (B, ZZ/k).
Combined with Theorem 6.1 of [DG], yields the following lemma.
Lemma 5.15. Let A = P Ml (C(X))P, and X, one of {pt}, [0, 1], TII,k , TIII,k
or S 2 , and let B be any C ∗ -algebra. Let φ, ψ ∈ Hom(A, B). Suppose that the
following statements hold.
1. When X = {pt}, [0, 1] or S 2 ,
[φ]∗ = [ψ]∗ : K0 (A) −→ K0 (B);
2. When X = TII,k ,
[φ]∗ = [ψ]∗ : K0 (A) −→ K0 (B)
and
[φ]∗ = [ψ]∗ : K1 (A, ZZ/k) −→ K1 (B, ZZ/k);
3. When X = TIII,k ,
[φ]∗ = [ψ]∗ : K0 (A) −→ K0 (B)
and
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
427
[φ]∗ = [ψ]∗ : K0 (A, ZZ/k) −→ K0 (B, ZZ/k).
It follows that, for any finite set F ⊂ A and any number ε > 0, there exist
n ∈ IN, µ ∈ Hom(A, Mn (B)) with finite dimensional image and a unitary
u ∈ Mn+1 (B) such that
ku(φ(a) ⊕ µ(a))u∗ − ψ(a) ⊕ µ(a)k < ε
for all a ∈ F .
5.16. Fix A = P Ml (C(X))P , X = {pt}, [0, 1], TII,k , TIII,k or S 2 . Then A is
stably isomorphic to C(X). By 5.14, an element α ∈ KK(A, B) is completely
determined by
α00 : K0 (A) → K0 (B),
αk0 : K0 (A, ZZ/k) → K0 (B, ZZ/k),
and
αk1 : K1 (A, ZZ/k) → K1 (B, ZZ/k).
Note that, for any C ∗ -algebra A,
K0 (A ⊗ C(Wk × S 1 )) ∼
= K0 (A) ⊕ K1 (A) ⊕ K0 (A, ZZ/k) ⊕ K1 (A, ZZ/k).
Each projection p ∈ M∞ (A ⊗ C(Wk × S 1 )) defines an element
[p] ∈ K0 (A) ⊕ K1 (A) ⊕ K0 (A, ZZ/k) ⊕ K1 (A, ZZ/k) ⊂ K(A).
S∞
This defines a map from the set of projections in k=2 M∞ (A ⊗ C(Wk × S 1 ))
to K(A).
S∞
For any finite set P ⊂ k=2 M∞ (A ⊗ C(Wk × S 1 )) of projections, denoted
by PK(A) the finite subset of K(A) consisting of elements coming from the
projections p ∈ P, that is
PK(A) = {[p] ∈ K(A)| p ∈ P}.
In particular, if A = P Ml (C(X))P, X = TII,k , or TIII,k , then we can choose
a finite set of projections PA ⊂ M• (A ⊗ C(Wk × S 1 )) such that the set {[p] ∈
K0 (A) ⊕ K1 (A) ⊕ K0 (A, ZZ/k) ⊕ K1 (A, ZZ/k) | p ∈ PA } = PA K(A) generates
K0 (A) ⊕ K1 (A) ⊕ K0 (A, ZZ/k) ⊕ K1 (A, ZZ/k) ⊂ K(A). For X = {pt}, [0, 1]
or S 2 , choose PA ⊂ M• (A) such that {[p] ∈ K0 (A) | p ∈ PA } generates
K0 (A) ⊂ K(A). We will use P to denote PA if there is no danger of confusion.
5.17. Let A = P Ml (C(X))P, X = {pt}, [0, 1], TII,k , TIII,k or S 2 , and P ⊂
M• (A ⊗ C(Wk × S 1 )) or P ⊂ M• (A) be as in 5.16. There are a finite subset
G(P) ⊂ A and a number δ(P) > 0 such that if B is any C ∗ -algebra and
φ ∈ Map(A, B) is G(P) − δ(P) multiplicative, then
k((φ ⊗ id)(p))2 − (φ ⊗ id)(p)k <
1
, ∀p ∈ P,
4
Documenta Mathematica 7 (2002) 255–461
428
Guihua Gong
| ). Hence for any
where id is the identity map on M• (C(Wk × S 1 )) or on M• (C
p ∈ P, there is a projection q ∈ M• (B ⊗ C(Wk × S 1 )) (or q ∈ M• (B) ) such
that
1
k(φ ⊗ id)(p) − qk < .
2
So q defines an element in K(B). (If q ′ is another projection satisfying the
same condition, then kq − q ′ k < 1, hence q ′ is unitarily equivalent to q.)
Therefore, if φ is G(P) − δ(P) multiplicative, then it induces a map
φ∗ : PK(A) → K(B).
Note that such G(P)
and δ(P) could be defined for any finite set P ⊂ M∞ (A)∪
S∞
M∞ (A ⊗ C(S 1 )) ∪ k=2 M∞ (A ⊗ C(Wk × S 1 )) of projections.
Theorem 5.18. Let X be one of the spaces {pt}, [0, 1], TII,k , TIII,k or S 2 .
Let A = P Ml (C(X))P and P be as in 5.16. For any finite set F ⊂ A, any
positive number ε > 0, and any positive integer M , there are a finite set G ⊂ A
(G ⊃ G(P) large enough ), a positive number δ > 0 (δ < δ(P) small enough),
and a positive integer L (large enough) such that the following statement is
true.
If φ, ψ ∈ Map(A, B) are G-δ multiplicative and
φ∗ = ψ∗ : PK(A) −→ K(B),
where B = QM• (C(Y ))Q with dim(Y ) ≤ M , then there is a homomorphism
ν ∈ Hom(A, ML (B)), with finite dimensional image, and there is a unitary
u ∈ ML+1 (B) such that
ku(φ ⊕ ν)(a)u∗ − (ψ ⊕ ν)(a)k < ε, ∀f ∈ F.
Proof: We first prove the theorem for the case B = M• (C(Y )). Then we apply
Lemma 1.3.6 to reduce the general case to this special case.
We prove the theorem by contradiction.
Let G(P) ⊂ G1 ⊂ G2 ⊂ · · · ⊂ Gn ⊂ · · · be a sequence of finite subsets with
∪Gn = unit ball of A.
Let δ(P) > δ1 > δ2 > · · · > δn > · · · be a sequence of positive numbers with
δn → 0. Let L1 < L2 < · · · < Ln < · · · be a sequence of positive integers with
Ln → +∞.
Suppose that the theorem does not hold for (Gn , δn , Ln ). That is, there exist
a C ∗ -algebra Bn = Mkn (C(Yn )) and two Gn -δn multiplicative maps
φn , ψn : A −→ Bn
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
429
with (φn )∗ = (ψn )∗ : PK(A) −→ K(Bn ) and
inf sup ku(φn ⊕ ν)(a)u∗ − (ψn ⊕ ν)(a)k ≥ ε,
ν,u a∈F
(∗)
where ν runs over all subsets of Hom(A, MLn (Bn )) consisting of those homomorphisms with finite dimensional images, and u runs over U (MLn +1 (Bn )).
+∞
The above {φn }+∞
n=1 , {ψn }n=1 induce two homomorphisms
φ̃, ψ̃ : A −→
+∞
Y
n=1
Bn
+∞
.M
Bn = Q(B).
n=1
We will prove that KK(φ) = KK(ψ).
1. X = {pt}, [0, 1] or S 2 . By Lemma 5.14, KK(φ̃) is completely determined by
([φ̃]∗ )00 : K0 (A) −→ K0 (Q(B)).
From 5.12,
+∞
.M
K0 (Q(B)) = Πb K0 (Bn )
K0 (Bn ).
n=1
([φ̃]∗ )00
That is, the above
is completely determined by the component ([φn ]∗ )00 .
From the condition that
[φn ]∗ = [ψn ]∗ : PK(A) −→ K(Bn )
and the condition that the group generated by PK(A) is K0 (A), we know that
KK(φ) = KK(ψ).
2. X = TII,k . By Lemma 5.14, KK(φ̃) is completely determined by
([φ̃]∗ )00 : K0 (A) −→ K0 (Q(B))
and
([φ̃]∗ )1k : K1 (A, ZZ/k) −→ K1 (Q(B), ZZ/k).
Furthermore, by 5.12,
K1 (Q(B), ZZ/k) =
+∞
Y
n=1
K1 (Bn , ZZ/k)
+∞
.M
K1 (Bn , ZZ/k).
n=1
Again, ([φ̃]∗ )00 and ([φ̃]∗ )1k are completely determined by the components corresponding to [φn ]∗ . And from [φn ]∗ = [ψn ]∗ on PK(A), we obtain KK(φ̃) =
KK(ψ̃). (Note that, we also use the fact that PK(A) generates a subgroup of
K(A) containing K0 (A) and K1 (A, ZZ/k).) (The subgroup of K(A) generated
by PK(A) also contains K1 (A), though we do not use this fact.)
3. X = TIII,k . It can be proved that KK(φ̃) = KK(ψ̃) as above. Note that
Q+∞
L+∞
K0 (Bn , ZZ/k) = n=1 K0 (Bn , ZZ/k)/ n=1 K0 (Bn , ZZ/k), by 5.12.
Documenta Mathematica 7 (2002) 255–461
430
Guihua Gong
By LemmaQ5.15, there
are a positive integer L and a homomorphism ν̃ :
L+∞
+∞
A → ML ( n=1 Bn / n=1 Bn ) with finite dimensional image, and a unitary
Q+∞
L+∞
ũ ∈ ML+1 ( n=1 Bn / n=1 Bn ), such that
kũ(φ̃ ⊕ ν̃)(a)ũ∗ − (ψ̃ ⊕ ν̃)(a)k <
ε
2
for all a ∈ F . Since ν̃ has finite dimensional image, one can find a sequence of
homomorphisms
νn : A −→ ML (Bn )
of finite dimensional images such that {νn }+∞
n=1 induces ν̃. One can also lift ũ
to a sequence of unitaries un ∈ ML+1 (Bn ). Then if n is large enough, we have
kun (φn ⊕ νn )(a)u∗n − (ψn ⊕ νn )(a)k < ε
for all a ∈ F . This contradicts with (∗) if one choose n to satisfy Ln ≥ L.
Now we apply Lemma 1.3.6 to prove the general case. Let G, δ and L1 (in
place of L) be as above for the case of full matrix algebras over C(Y ) with
dim(Y ) ≤ M . Choose L = (2M + 2)L1 − 1. We will verify that G, δ and
L satisfies the condition of the theorem even for B = QM• (C(Y ))Q—cutting
down of full matrix algebras by projections, as follows.
Let n = rank(Q) + dim(Y ) and m = 2M + 1. Then by Lemma 1.3.6,
QM• (C(Y ))Q can be identified as a corner subalgebra of Mn (C(Y )), and
Mn (C(Y )) can be identified as corner subalgebra of Mm (QM• (C(Y ))Q).
If φ, ψ ∈ MapG−δ (A, B) satisfy the condition in the theorem, then regarding
B as a corner subalgebra of Mn (C(Y )), we can regard φ, ψ as elements in
MapG−δ (A, Mn (C(Y ))) which still satisfy the condition. Hence from the above
special case of the theorem, there are ν : A → ML1 (Mn (C(Y ))) and a unitary
u1 ∈ ML1 +1 (Mn (C(Y ))) such that
ku1 (φ ⊕ ν)(a)u∗1 − (ψ ⊕ ν)(a)k < ε, ∀a ∈ F.
Also, Mn (C(Y )) can be regarded as a corner subalgebra of Mm (QM• (C(Y ))Q),
so φ ⊕ ν and ψ ⊕ ν can be regarded as maps from A to
ML1 +1 (Mm (QM• (C(Y ))Q)) = ML+1 (QM• (C(Y ))Q). Therefore, there is
a unitary u ∈ ML+1 (B)
ku(φ ⊕ ν)(a)u∗ − (ψ ⊕ ν)(a)k < ε, ∀a ∈ F.
⊔
⊓
Remark 5.19. The theorem is not true for X = S 1 , even if we assume that
both φ and ψ are homomorphisms. A counterexample is given below .
Let φn , ψn : C(S 1 ) → C[0, 1] be defined by
φn (f )(t) = f (e2πint )
and
ψn (f )(t) = f (1).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
431
Then KK(φn ) = KK(ψn ). Let F = {z} and ε = 41 , where z ∈ C(S 1 ) is a
canonical generator. One can prove that there is no integer L which is good
for all (φn , ψn ) as in Theorem 5.18, by using the variation of determinant.
5.20. If X is any finite CW complex such that K1 (C(X)) is a torsion group,
then Theorem 5.18 holds for X — one needs to choose PA accordingly, which
is described below.
Suppose that m1 , m2 , · · · , mi are the degrees of all the torsion elements in
K0 (A) and K1 (A). Let m be the least common multiple of m1 , m2 , · · · , mi .
Since K1 (A) is a torsion group, similar to the discussion in 5.13, an element
α ∈ KK(A, B) is completely determined by the K-theory maps
α0 : K0 (A) −→ K0 (B),
αp0 : K0 (A, ZZ/p) −→ K0 (B, ZZ/p),
αp1 : K1 (A, ZZ/p) −→ K1 (B, ZZ/p),
where p are all the numbers with p|m. (In particular, the map α1 : K1 (A) −→
K1 (B) is determined by the above maps.)
One S
can
S choose P to be1 a finite set of projections in
M• (A)
p|m M• (A ⊗ C(Wp ⊗ S )) such that the set PK(A) (defined in
5.16) generates a sub group containing the group
M
K∗ (A, ZZ/k).
K0 (A) ⊕
k|m
Similar to the proof of Theorem 5.18, we can prove the following theorem,
since, to determine a KK-element α ∈ KK(A, B), one does not need the map
from K1 (A). (G(P) and δ(P) can be chosen accordingly as in 5.17.)
Theorem 5.21. Suppose that X is a finite CW complex with K1 (X) a torsion
group. Suppose that A = P Ml (C(X))P and P are as in 5.20. For any finite
set F ⊂ A, positive number ε > 0, and positive integer M , there are a finite
set G ⊂ A (G ⊃ G(P) large enough ), a positive number δ > 0 (δ < δ(P)
small enough), and a positive integer L (large enough) such that the following
statement is true.
If φ, ψ ∈ Map(A, B) are G(P)-δ(P) multiplicative and
φ∗ = ψ∗ : PK(A) −→ K(B),
where B = QM• (C(Y ))Q with dim(Y ) ≤ M , then there is a homomorphism
ν ∈ Hom(A, ML (B)) with finite dimensional image, and there is a unitary
u ∈ ML+1 (B) such that
ku(φ ⊕ ν)(a)u∗ − (ψ ⊕ ν)(a)k < ε
for all a ∈ F .
Documenta Mathematica 7 (2002) 255–461
432
Guihua Gong
The following is a direct consequence of Theorem 5.18.
Corollary 5.22.
Let A = C(X), where X is one of the spaces:
[0, 1], S 2 , TII,k or TIII,k , and let P be as in 5.16. For any finite set F ⊂ C(X),
any positive number ε > 0 and any positive integer M , there are a finite set
G ⊂ C(X) (G ⊃ G(P) large enough), positive numbers δ > 0 (δ ≤ δ(P) small
enough) and η > 0 (small enough) such that the following statement is true.
Let B = M• (C(Y )) with dim(Y ) ≤ M , and p ∈ B, a projection.
If φ, ψ ∈ Map(C(X), pBp) are G-δ multiplicative maps inducing the same maps
φ∗ = ψ∗ : PK(C(X)) −→ K(B), and {x1 , x2 , · · · , xn } is an η-dense subset
of X, and q1 , q2 , · · · , qn are mutually orthogonal projections in (1 − p)B(1 − p)
with rank(qi ) ≥ rank(p), then there is a unitary
u ∈ (p ⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )B(p ⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )
such that
°
! °
Ã
n
n
°
°
X
X
°
°
f (xi )qi u∗ ° < ε,
f (xi )qi − u ψ(f ) ⊕
°φ(f ) ⊕
°
°
i=1
i=1
∀f ∈ F.
In particular, if ψ is a homomorphism, then there is a homomorphism φ̃ ∈
Hom(C(X),
⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )B(p ⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )) (defined by φ̃(f ) =
P(p
n
u (ψ(f ) ⊕ i=1 f (xi )qi ) u∗ ) such that
°
!°
Ã
n
°
°
X
°
°
f (xi )qi ° < ε,
∀ f ∈ F.
°φ̃(f ) − φ(f ) ⊕
°
°
i=1
Proof: Since X is not the space of a single point, we can assume that X,
as a metric space, satisfies that diameter(X) = 1. Apply Theorem 5.18 to
the finite set F ⊂ A, the positive number 3ε and the integer M to obtain
G, δ, L as in Theorem 5.18. Choose a positive number η < 8M1L2 such that if
dist(x, x′ ) < 8M L2 · η, then kf (x) − f (x′ )k < 3ε for all f ∈ F .
Let {x1 , x2 , · · · , xn } be an η-dense subset of X and let q1 , q2 , · · · , qn ∈ (1 −
p)B(1 − p) be mutually orthogonal projections with rank(qi ) ≥ rank(p). Similar to the proof of Corollary 1.6.13, one can find a 8M L · η-dense subset {xk1 , xk2 , · · · , xkl } ⊂ {x1 , x2 , · · · , xn } and mutually orthogonal projections
Q1 , Q2 , · · · , Ql with rank(Qj ) ≥ M L · rank(p), such that
k
n
X
i=1
f (xi )qi −
l
X
j=1
f (xkj )Qj k <
ε
, ∀f ∈ F.
3
Since rank(Qi ) ≥ M L · rank(p), it follows that [Qi ] ≥ L · [p].
Again, similar to the proofs of Corollaries 1.6.12 and 1.6.13, it can be proved
that a homomorphism ν ∈ Hom(A, ML (pBp)) with finite dimensional image
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
(from 5.18) can be perturbed, at the expense of at most
to a homomorphism ν ′ which is of the form
ν ′ (f ) =
l
X
ε
3
433
on the finite set F ,
f (xkj )qj′
j=1
with [qj′ ] ≤ [Qj ] (some of the projections qj′ could be zero). Hence the
corollary follows (see the proof of Corollary 1.6.12).
⊔
⊓
By the discussion in 1.2.19, we have the following corollary.
Corollary 5.23.
Let A = Ml (C(X)), where X is one of the spaces:
[0, 1], S 2 , TII,k or TIII,k , and let P be as in 5.16. For any finite set F ⊂ A,
any positive number ε > 0 and any positive integer M , there are a finite set
G ⊂ A (G ⊃ G(P) large enough), numbers δ > 0 (δ ≤ δ(P) small enough) and
η > 0 (small enough) such that the following statement is true.
Let B = M• (C(Y )) with dim(Y ) ≤ M , and p ∈ B a projection.
If φ, ψ ∈ Map(A, pBp) are G-δ multiplicative maps inducing the same map
φ∗ = ψ∗ : PK(A) −→ K(B), and {x1 , x2 , · · · , xn } is an η-dense subset of X,
and
q1 = q1′ ⊕ q1′ ⊕ · · · ⊕ q1′ , q2 = q2′ ⊕ q2′ ⊕ · · · ⊕ q2′ , · · · , qn = qn′ ⊕ qn′ ⊕ · · · ⊕ qn′
{z
}
{z
}
{z
}
|
|
|
l
l
l
are mutually orthogonal projections in (1 − p)B(1 − p) with rank(qi ) ≥ rank(p),
then there is a unitary
u ∈ (p ⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )B(p ⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )
such that
°
! °
Ã
n
n
°
°
X
X
°
°
qi′ ⊗ f (xi ) u∗ ° < ε,
qi′ ⊗ f (xi ) − u ψ(f ) ⊕
°φ(f ) ⊕
°
°
i=1
i=1
∀f ∈ F.
In particular, if ψ is a homomorphism, then there is a homomorphism φ̃ ∈
Hom(C(X), (p ⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )B(p ⊕ q1 ⊕ q2 ⊕ · · · ⊕ qn )) such that
°
!°
Ã
n
°
°
X
°
°
′
qi ⊗ f (xi ) ° < ε,
∀ f ∈ F.
°φ̃(f ) − φ(f ) ⊕
°
°
i=1
Proof: Thanks to Lemma 1.6.8, we can always assume that the two maps φ and
ψ satisfy the condition that φ|Ml (C
| ) and ψ|Ml (C
| ) are homomorphisms. Using
the condition φ∗ = ψ∗ : PK(A) −→ K(B), we can assume
φ|Ml (C
| ) = ψ|Ml (C
| )
Documenta Mathematica 7 (2002) 255–461
434
Guihua Gong
after conjugating with a unit.
Now the corollary follows from the following claim.
Claim: For any finite set F ⊂ A = Ml (C(X)), any ε > 0, there are a finite set
G ⊂ A and a positive number δ > 0 such that if a map φ : A → B is G-δ multiplicative and φ|Ml (C
| ) is a homomorphism, then there are a map φ1 : C(X) →
φ(e11 )Bφ(e11 ) and an identification of φ(1)Bφ(1) ∼
= Ml (φ(e11 )Bφ(e11 )) such
that
kφ(f ) − (φ1 ⊗ 1l )(f )k < ε
∀f ∈ F.
Furthermore, if G1 ⊂ C(X) and δ1 > 0 are a pregiven finite set and a pregiven
positive number, then one can modify the set G and the number δ so that the
map φ1 above can be chosen to be G1 -δ1 multiplicative.
Proof of Claim: Suppose 1 ∈ F . Let F1 = {aij |(aij )l×l ∈ F
P}(⊂ C(X)) be the
set of all entries of the elements in¡F . Let¢ G = {(bij )l×l = bij eij | bij ∈ F1 ∪
G1 ⊂ C(X)}( ⊂ A) and δ = min 2lε2 , δ1 . Suppose that φ : Ml (C(X)) → B
is G-δ multiplicative. Let
φ1 = φ|e11 Ml (C(X))e11 : C(X) → φ(e11 )Bφ(e11 ).
Obviously the G-δ multiplicativity of φ implies the G1 -δ1 multiplicativity of φ1 .
Identify φ(1)Bφ(1) ∼
= (φ(e11 )Bφ(e11 )) ⊗ Ml by sending φ(eij ) to eij ∈ Ml ⊂
(φ(e11 )Bφ(e11 )) ⊗ Ml . Under this identification, we have
X
(φ1 ⊗ 1l )(a) =
φ(ei1 )φ1 (aij )φ(e1j ),
i,j
where
a = (aij )l×l ∈ Ml (C(X)). On the other hand, writing a =
P
e1i (aij e11 )e1j and using the G-δ multiplicativity of φ, we have
X
kφ(a) − (φ1 ⊗ 1l )(a)k ≤
kφ(e1i (aij e11 )e1j ) − φ(ei1 )φ1 (aij )φ(e1j )k
i,j
=
X
i,j
≤
kφ(e1i (aij e11 )e1j ) − φ(ei1 )φ(aij e11 )φ(e1j )k
X
i,j
2δ = 2l2 δ ≤ ε.
This proves the Claim.
Applying the Claim, one can reduce the proof to the case A = C(X) which is
Corollary 5.22.
⊔
⊓
Definition 5.24. Let A be a unital C ∗ -algebra, let
P ⊂ M• (A) ∪ M• (A ⊗ C(S 1 )) ∪
∞
[
k=2
M• (A ⊗ C(Wk × S 1 ))
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
435
be a finite set of projections, and let G(P), δ(P) be as in 5.17. A G(P) − δ(P)
multiplicative map φ : A → B is called quasi-PK-homomorphism if there is
a homomorphism ψ : A → B with φ(1A ) = ψ(1A ) such that
[φ]∗ = [ψ]∗ : PK(A) → K(B).
Using the above definition and Definition 4.38, we can restate the second part
of Corollary 5.23 as below.
Lemma 5.25.
Let A = Ml (C(X)), where X is one of the spaces:
[0, 1], S 2 , TII,k or TIII,k , and let P be as in 5.16. For any finite set F ⊂ A,
any positive number ε > 0 and any positive integer M , there are a finite set
G ⊂ A (G ⊃ G(P) large enough), positive numbers δ > 0 (δ ≤ δ(P) small
enough) and η > 0 (small enough) such that the following statement is true.
Let B = M• (C(Y )) with dim(Y ) ≤ M , and let p ∈ B be a projection. If
φ ∈ Map(A, pBp) is a G-δ multiplicative quasi-PK-homomorphism, and λ ∈
Hom(A, (1 − p)B(1 − B)) has the property PE(rank(p), η), then there is a
homomorphism φ̃ ∈ Hom(A, B) such that
kφ̃(f ) − (φ ⊕ λ)(f )k < ε,
∀f ∈ F.
Furthermore if Y is a connected simplicial complex different from the single
point space, then φ̃ can be chosen to be injective.
Proof: The main body of the lemma is a restatement of Corollary 5.23. So
we only need to prove the last sentence of the lemma. We need the following
fact: Let X = [0, 1], S 2 , TII,k or TIII,k , and let Y be a connected finite simplicial complex different from {pt}. If λ1 : Ml (C(X)) → p1 M• (C(Y ))p1 is a
homomorphism defined by the point evaluation at a point x1 ∈ X as
ex
1
| ) −→ p1 M• (C(Y ))p1 ,
Ml (C
Ml (C(X)) −→
then λ1 is homotopic to an injective homomorphism λ′1 : Ml (C(X)) →
p1 M• (C(Y ))p1 . (Again, this fact can be proved by using the Peano Curve.)
Let η ′ be as the η desired in the main body of the lemma for 2ε (in place of
ε). We can also assume that η ′ satisfies the condition that if dist(x, x′ ) < η ′ ,
′
then kf (x) − f (x′ )k < 2ε for all f ∈ F . Choose η = η4 . Suppose that
Ln λ ∈
Hom(A, (1 − p)B(1 − p)) has the property PE(rank(p), η). Write λ = i=1 λi ,
where
exi
φi
| ) −→
pi Bpi ,
Ml (C
λi : Ml (C(X)) −→
are point evaluations at an η-dense set {x1 , x2 , · · · , xn } and φi are unital homomorphisms.
Let p1 be a projection with minimum rank among all the projections
p1 , p2 , · · · , pn . Let φ ∈ Map(A, pBp) be a G-δ multiplicative quasi-PKhomomorphism. Then φ ⊕ λ1 ∈ Map(A, (p ⊕ p1 )B(p ⊕ p1 )) is also a G-δ
Documenta Mathematica 7 (2002) 255–461
436
Guihua Gong
multiplicative quasi-PK-homomorphism. Furthermore, from the above fact,
it defines the same map on the level of PK(A) as an injective homomorLn
phism ψ ∈ Hom(A, (p ⊕ p1 )B(p ⊕ p1 )). On the other hand, λ′ = Li=2 λi
n
has the properties PE(rank(p), 2η) and PE(rank(p1 ), 2η), since λ =
i=1 λi
has PE(rank(p), η), and rank(p1 ) ≤ rank(pi ), i = 2, · · · , n. Similar to the
proof of Corollary 5.22, λ′ can be perturbed to a homomorphism λ′′ which
has the property PE(rank(p) + rank(p1 ), 4η) at the expense of at most 2ε on
the finite set F . Note that 4η = η ′ , and ψ is injective. Hence the homomorphism Adu ◦ (ψ ⊕ λ′′ ) (for a certain unitary u), as desired in the main body
of the lemma, is also injective.
⊔
⊓
Lemma 5.26. Let X and Y be connected finite simplicial complexes. Suppose that φ1 : P Mk (C(X))P → Q1 Ml (C(Y ))Q1 and φ2 : P Mk (C(X))P →
Q2 Ml (C(Y ))Q2 are unital homomorphisms, where P, Q1 , and Q2 are projections with
rank(Q2 ) − rank(Q1 ) ≥ 2 dim(Y ) · rank(P ).
Then there exists a homomorphism ψ : P Mk (C(X))P → M• (C(Y )) such that
[ψ] = [φ2 ] − [φ1 ] ∈ KK(C(X), C(Y )).
Proof: First, we suppose that A = C(X). As in Lemma 3.14 of [EG 2]
(see Remark 1.6.21 above), we can assume that φ1 (C0 (X)) ⊂ Ml (C0 (Y )) and
φ2 (C0 (X)) ⊂ Ml (C0 (Y )), where C0 (X) and C0 (Y ) are sets of functions vanishing on fixed base points of X and Y , respectively. Hence φi defines an
element kk(φi ) ∈ kk(Y, X) (see [DN]). Furthermore, [φi ] ∈ KK(C(X), C(Y ))
is completely determined by kk(φi ) and φi∗ ([1A ]) ∈ K0 (B). Let α = kk(φ2 ) −
kk(φ1 ) ∈ kk(Y, X) (note that kk(Y, X) is an abelian group, see [DN]). Since
rank(Q2 )−rank(Q1 ) ≥ 2 dim(Y ), by [Hu], there is a projection Q3 ∈ M• (C(Y ))
such that [Q3 ] = [Q2 ] − [Q1 ] ∈ K0 (C(Y )). By Theorem 4.11 of [DN] or Lemma
3.16 of [EG2], there is a unital homomorphism ψ : C(X) → Q3 M• (C(Y ))Q3
to realize α ∈ kk(Y, X). Obviously ψ is as desired.
For the general case, using the Dilation Lemma (Lemma 1.3.1), one can prove
that [φi ] ∈ KK(C(X), C(Y )) can be realized by homomorphism φ′i : C(X) →
M• (C(Y )). This reduces the proof to the above case.
⊔
⊓
Remark 5.27. In the above lemma, if Q1 < Q2 , then one can choose ψ to
satisfy ψ(1A ) = Q2 − Q1 .
Lemma 5.28.
Let X be a finite simplicial complex, and A = P Ml (C(X))P .
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
437
For any finite set
P ⊂ M• (A) ∪ M• (A ⊗ C(S 1 )) ∪
∞
[
k=2
M• (A ⊗ C(Wk × S 1 )),
there are a finite set G ⊂ A and a number δ > 0, such that the following is
true.
If Y is a simplicial complex, Q > Q1 are two projections in M• (C(Y )) with
rank(Q) − rank(Q1 ) ≥ 2 dim(Y )rank(P ), and two unital homomorphisms φ ∈
Hom(A, QM• (C(Y ))Q)1 , φ1 ∈ Hom(A, Q1 M• (C(Y ))Q1 )1 and a unital map
φ2 ∈ Map(A, (Q − Q1 )M• (C(Y ))(Q − Q1 ))1 satisfy that
(∗)
kφ(f ) − φ1 (f ) ⊕ φ2 (f )k < δ, ∀g ∈ G,
then there is a homomorphism ψ : A → (Q − Q1 )M• (C(Y ))(Q − Q1 ) such that
[ψ]∗ = [φ2 ]∗ : PK(A) → K(C(Y )).
In other words, φ2 is a quasi-PK-homomorphism.
(Notice that, from Lemma 4.40, if G is large enough and δ is small enough,
then (*) above implies that
φ2 ∈ Map(A, (Q − Q1 )M• (C(Y ))(Q − Q1 ))1 is G(P) − δ(P) multiplicative, and
hence [φ2 ]∗ : PK(A) → K(C(Y )) makes sense.)
Proof: If G is large enough and δ is small enough, then (*) implies
[φ2 ]∗ = [φ]∗ − [φ1 ]∗ : PK(A) → K(C(Y )).
Then the lemma follows from Lemma 5.26 and Remark 5.27.
⊔
⊓
Remark 5.29. In Corollary 4.39, we can choose ψ0 (or ψ0′ ) such that ψ0i,j (or
′
ψ0i,j ) is a quasi-PK-homomorphism for any pre-given set of projections
1
P ⊂ M• (A) ∪ M• (A ⊗ C(S )) ∪
∞
[
k=2
M• (A ⊗ C(Wk × S 1 )).
To do so, by Lemma 5.28, one only needs to choose the projection Qi,j
0 to have
rank at least 2 dim(Xm,j ) · rank(1Ain ). But from the construction in 4.34, we
have freedom to do so.
Ls
Lemma 5.30. Fix a positive integer M . Suppose that B = i=1 Mli (C(Yi )),
where Yi are the spaces: {pt}, [0, 1], S 1 , TII,k , TIII,k , S 2 . For any finite set G ⊂
B and positive number ε > 0, there exist a finite set G1 ⊂ B, numbers δ1 > 0
and η > 0 such that the following is true.
Lt
If a map α = α0 ⊕ α1 : B → A =
j=1 Mkj (C(Xj )), with dim(Xj ) ≤ M ,
satisfies the following conditions:
Documenta Mathematica 7 (2002) 255–461
438
Guihua Gong
(1) α0 is G1 -δ1 multiplicative, {α0 (1B i )}si=1 are mutually orthogonal projections, and α1 is a homomorphism with finite dimensional image (i.e., defined
by point evaluations);
(2) For any block B i with Yi = TII,k , TIII,k or S 2 and any block Aj , the
partial map α0i,j is quasi-PK-homomorphism, where P is the set of projections associated to B i as in 5.16, and the homomorphism α1i,j has the property
PE(rankα0i,j (1B i ), η);
then there is a unital homomorphism α′ : B → α(1B )Aα(1B ) such that
kα′ (g) − α(g)k < ε, ∀g ∈ G.
Proof: We only need to perturb all the individual maps αi,j to homomorphisms
′
α i,j within αi,j (1B i )Aj αi,j (1B i ).
For a block of B i with spectrum {pt}, [0, 1] or S 1 , such perturbation exists
by Lemma 1.6.1. For a block of B i with spectrum TII,k , TIII,k or S 2 , such
perturbation exists by Lemma 5.25.
⊔
⊓
Lemma 5.31. Let M be a fixed positive integer. Let B = Ml (C(Y )), Y =
TII,k , TIII,k or S 2 . Let the set of projections P ⊂ M• (B)∪M• (B ⊗C(Wk ×S 1 ))
be as in 5.16.
Let A = RMl1 (C(X))R with dim(X) ≤ M , where R ∈ Ml1 (C(X)) is a projection. Let α : B → A be an injective homomorphism. Let a finite set of
projections P ′ be given by P ′ := (α ⊗ id)(P) ⊂ M• (A) ∪ M• (A ⊗ C(Wk × S 1 )).
Let η > 0. Choose ηS
1 > 0 such that if a finite set {x1 , x2 , · · · , xn } ⊂ X is
n
η1 -dense in X, then i=1 SPαxi is η-dense in Y . (Such η1 exists because of
injectivity of α.)
For any finite subset G1 ⊂ B and any number δ1 > 0, there are a finite subset
G2 ⊂ A and a number δ2 > 0 such that the following are true.
Let C = M• (Z) with dim(Z) ≤ M .
(1) If ψ0 : A → Q0 CQ0 is a G2 -δ2 multiplicative quasi-P ′ K-homomorphism
and ψ0 (α(1B )) is a projection, then ψ0 ◦ α is a G1 -δ1 multiplicative quasi-PKhomomorphism.
(2) If ψ1 : A → Q1 CQ1 has the property PE(J · L, η1 ), where J = rank(R),
then ψ1 ◦ α : B → ψ1 (α(1B ))Cψ1 (α(1B )) has property PE(L, η).
In particular, if ψ1 has the property PE(J · rank(Q0 ), η1 ), (this is the condition
(3) of Corollary 4.39), then ψ1 ◦ α has the property PE(rank((ψ0 ◦ α)(1B )), η).
(Note that rank((ψ0 ◦α)(1B )) ≤ rank(Q0 ).) Consequently, if we further assume
that G1 , δ1 and η are as those chosen in 5.30 for a finite set G ⊂ B and
ε > 0, and Q1 is orthogonal to Q0 , then there is a homomorphism ψ : B →
(Q0 ⊕ Q1 )C(Q0 ⊕ Q1 ) such that
kψ(g) − (ψ0 ⊕ ψ1 )(α(g))k < ε, ∀g ∈ G.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
439
Proof: (1) holds if we choose G2 ⊃ α(G) and δ2 < δ1 .
(2) follows from the following fact: if a homomorphism φ : RMl1 (C(X))R →
Mt (C(Z)) contains a part of point evaluation at a point x ∈ X of size at least
L (see Definition 4.38), then for any y ∈ SPαx ⊂ Y , φ ◦ α contains a part of
L
point evaluation at point y of size at least
.
rank(R)
⊔
⊓
The following two theorems are important for the proof of our main theorem.
Theorem 5.32a.
Let M be a positive integer.
Let
Lk n
M
(C(X
)),
φ
)
be
a
simple
inductive
limit
with
lim (An =
n,i
n,m
[n,i]
i=1
n→∞
injective connecting
Ls homomorphisms φn,m and with dim(Xn,i ) ≤ M, for any
n, i. Let B = i=1 Mli (C(Yi )), where Yi are the spaces: {pt}, [0, 1], S 1 , TII,k ,
TIII,k , and S 2 .
Suppose that a homomorphism α : B → An satisfies the following dichotomy
condition:
For any block B i of B and any block Ajn of An , either the partial map αi,j :
it has a finite dimensional
image.
B i → Ajn is injective or L
L i
Denote α(1B ) := R(=
Ri ) ∈ An (=
An ). For any finite sets G ⊂ B
and F ⊂ RAn R, any positive number ε > 0, and any positive integer L,
there are Am and mutually orthogonal projections Q0 , Q1 , Q2 ∈ Am , with
φn,m (R) = Q0 + Q1 + Q2 , a unital map θ0 ∈ Map(RAn R, Q0 Am Q0 )1 , two
unital homomorphisms θ1 ∈ Hom(RAn R, Q1 Am Q1 )1 and
ξ ∈ Hom(RAn R, Q2 Am Q2 )1 such that
(1) kφn,m (f ) − (θ0 (f ) + θ1 (f ) + ξ(f ))k < ε, ∀f ∈ F ;
(2) there is a homomorphism α1 : B → (Q0 ⊕ Q1 )Am (Q0 ⊕ Q1 ) such that
kα1 (g) − (θ0 + θ1 ) ◦ α(g)k < ε, ∀g ∈ G;
(3) θ0 is F − ε multiplicative and θ1 satisfies that for any nonzero projection
(including any rank 1 projection) e ∈ Ri Ain Ri
θ1i,j ([e]) ≥ L · [θ0i,j (Ri )],
(the condition (3) will be used when we apply Theorem 1.6.9 in the proof of the
Main Theorem);
(4) ξ factors through a C ∗ -algebra C—a direct sum of matrix algebras over
| — as
C[0, 1] or C
ξ1
ξ2
ξ : RAn R −→ C −→ Q2 Am Q2 ,
and the partial maps of ξ2 satisfy the dichotomy condition;
(5) the partial maps of α1 satisfies the dichotomy condition.
Proof: Let E i,j = αi,j (1B i ) ∈ Ajn . Let
I = {(i, j) | αi,j : B i → Ajn has finite dimensional image}.
Documenta Mathematica 7 (2002) 255–461
440
Guihua Gong
L i
Let the subalgebra D ⊂ An =
An be defined by
M
M
M
M
| · 1 i ) ⊂
D=
αi,j (B i ) ⊕
αi,j (C
Ajn .
B
j
(i,j)∈I
(i,j)∈I
/
j
Notice that D is a finite dimensional subalgebra of An containing the mutually
orthogonal projections {E i,j = αi,j (1Bi )}i,j .
Apply part 2 of Corollary 4.39 for sufficiently large set F ′ ⊂ RAn R, sufficiently
small number ε′ > 0 and η ′ > 0, and positive integer J = L · maxi rank(Ri ), to
obtain Am and the decomposition θ0 ⊕θ1 ⊕ξ of φn,m |RAn R as ψ0′ ⊕ψ1′ ⊕ψ2′ in 4.39.
By Lemma 1.6.8, we can assume that the restriction θ0 |D is a homomorphism.
The condition (1) follows if we choose F ′ ⊃ F , and ε′ < ε.
The F − ε multiplicativity of θ0 in (3) follows from Lemma 4.40, if F ′ is large
enough and ε′ is small enough, and the desired property of θ1 in (3) follows
from the choice of J and Lemma 5.31.
To construct α1 as desired in the condition (2), we need to construct
α1i,j,k : B i → θj,k (E i,j )Akm θj,k (E i,j ),
where θ = θ0 ⊕ θ1 , to satisfy
kα1i,j,k (g) − θ j,k ◦ αi,j (g)k < ε, ∀g ∈ G.
The construction are divided into three cases.
1. If (i, j) ∈ I, then θ j,k ◦ αi,j is already a homomorphism and can be chosen
to be α1i,j,k .
2. If (i, j) ∈
/ I, and Yi = [0, 1] or S 1 , then the existence of α1i,j,k follows from
Lemma 1.6.1 and Lemma 4.40, if F ′ is large enough and ε′ is small enough.
(See Lemma 5.30 also.) In fact, in this case, the map θ0j,k ◦ αi,j itself can
be perturbed to a homomorphism. On the other hand, the homomorphism
θ1j,k ◦ αi,j is defined by the point evaluations on an η-dense set for a certain
small number η. Evidently, such a homomorphism θ1j,k ◦ αi,j from Mli (C(S 1 ))
or Mli (C([0, 1])) (to Akm ) can be perturbed to an injective homomorphism,
provided that η is sufficiently small and that the path connected simplicial
complex Xm,k is not the space of a single point. Therefore, in this case, the
homomorphism α1i,j,k can be chosen to be injective.
3. If (i, j) ∈
/ I, and Yi = TII,k , TIII,k or S 2 , then αi,j is injective, and the
existence of α1i,j,k follows from Lemma 5.30 and the choice of J, if F ′ is large
enough and ε′ is small enough, and if we choose η ′ to be the number η1 in
Lemma 5.31 corresponding to the η in Lemma 5.30. The homomorphism α1i,j,k
can also be chosen to be injective, if Xm,k is not the space of a single point,
according to the last part of Lemma 5.25.
L
Finally, define the partial map α1i,k of α1 to be j α1i,j,k to complete the construction. Obviously, it follows, from the discussion of the injectivity in case 2
and case 3, that α1 satisfies the dichotomy condition.
⊔
⊓
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
441
Theorem 5.32b.
Let M be a positive integer.
Let
Lk n
M
(C(X
)),
φ
)
be
a
simple
inductive
limit
with
lim (An =
n,i
n,m
[n,i]
i=1
n→∞
injective connecting
Ls homomorphisms φn,m and with dim(Xn,i ) ≤ M, 1for any
n, i. Let B =
i=1 Mli (C(Yi )), where Yi are the spaces:{pt}, [0, 1], S , TII,k ,
TIII,k , and S 2 .
Suppose that a homomorphism α : B → An satisfies the following dichotomy
condition:
For any block B i of B and any block Ajn of An , either the partial map αi,j :
B i → Ajn is injective or it has a finite dimensional image.
For any finite sets G ⊂ B and F ⊂ An , and any number ε > 0, there
are Am and mutually orthogonal projections P, Q ∈ Am , with φn,m (1An ) =
P + Q, a unital map θ ∈ Map(An , P Am P )1 , and a unital homomorphism
ξ ∈ Hom(An , QAm Q)1 such that
(1) kφn,m (f ) − (θ(f ) ⊕ ξ(f ))k < ε, ∀f ∈ F ;
(2) there is a homomorphism α1 : B → P Am P such that
kα1 (g) − (θ ◦ α)(g)k < ε, ∀g ∈ G;
(3) θ(F ) is weakly approximately constant to within ε;
(4) ξ factors through a C ∗ -algebra C—a direct sum of matrix algebras over
| — as
C[0, 1] or C
ξ2
ξ1
ξ : An −→ C −→ QAm Q,
and the partial maps of ξ2 satisfy the dichotomy condition;
(5) the partial maps of α1 satisfy the dichotomy condition.
The proof is similar to the proof of Theorem 5.32a, we omit it.
6
The proof of the main theorem
In this section, we will combine §4, §5 and §1.6 to prove our Main Theorem —
the Reduction Theorem.
The following is Proposition 3.1 of [D2].
Proposition 6.1. ([D2, 3.1]) Consider the diagram
Ax1
α1
B1
φ1,2
−→
β1
ց
ψ1,2
−→
Ax2
α2
B2
φ2,3
−→
β2
ց
ψ2,3
−→
···
···
···
−→ A
xn
αn
−→ Bn
φn,n+1
−→
βn
ց
ψn,n+1
−→
Axn+1
αn+1
Bn+1
−→ · · ·
ց
−→ · · ·
,
where An , Bn are C ∗ -algebras, φn,n+1 , ψn,n+1 are homomorphisms and αn , βn
are linear ∗-contractions.
Documenta Mathematica 7 (2002) 255–461
442
Guihua Gong
Suppose that Fn ⊂ An , En ⊂ Bn are finite sets satisfying the following conditions.
φn,n+1 (Fn ) ∪ αn+1 (En+1 ) ⊂ Fn+1 ,
ψn,n+1 (En ) ∪ βn (Fn ) ⊂ En+1 ,
S∞
S∞
and
n=1 (φn,∞ (Fn )) and
n=1 (ψn,∞ (En )) are the unit balls of A =
lim(An , φn,m ) and B = lim(Bn , ψn,m ), respectively.
Suppose that there is a
P
sequence ε1 , ε2 , · · · of positive numbers with
εn < +∞ such that αn and βn
are Fn -εn multiplicative and En -εn multiplicative, respectively, and
kφn,n+1 (f ) − αn+1 ◦ βn (f )k < εn ,
and
kψn,n+1 (g) − βn ◦ αn (g)k < εn
for all f ∈ Fn and g ∈ En .
Then A is isomorphic to B.
Ltn
M[n,i] (C(Xn,i )), φn,m ) be a simple inductive
Lemma 6.2. Let lim(An = i=1
limit C ∗ -algebra with φn,m injective, where Xn,i are path connected finite simplicial complexes with uniformly bounded dimensions. Let C ∗ -algebra C be a
direct sum of matrix algebras over the spaces: {pt}, [0, 1], S 1 , TII,k , TIII,k and
S 2 , and φ : C → An be an injective homomorphism. Then for any finite set
F ⊂ C and ε > 0, there is a positive integer N > n such that for any m > N ,
there is a homomorphism ψ : C → Am satisfying the following conditions.
(1) ψ(1C i ) = (φn,m ◦ φ)(1C i ), for any block C i of C.
(2) kψ(f ) − (φn,m ◦ φ)(f )k < ε, ∀f ∈ F.
(3) ψ satisfies the following dichotomy condition:
For any block C i of C and Ajm of Am , either ψ i,j is injective or ψ i,j has finite
dimensional image.
(For the proof of the main theorem of this article—Theorem 6.3 below, we only
need this lemma for the case that C is a direct sum of matrix algebras over
spaces {pt} and [0, 1]. The full generality of the lemma will be used in the
proof of Corollary 6.11 below.)
Proof: We only need to prove for the case that C has only one block C =
Mk (C(X)). And, by the discussion in 1.2.19, this case can further be reduced
to the case C = C(X).
For the finite set F ⊂ C, there is an η > 0 such that if dist(t, t′ ) < 4η, then
kf (t) − f (t′ )k < ε, ∀f ∈ F.
Let (X, σ) be a simplicial decomposition of X such that for any simplex ∆ ⊂
(X, σ), diameter(∆) < η. We call a simplex ∆ a top simplex if ∆ is not a proper
face of any simplex. Obviously, ∆ is a top simplex if and only if the interior
◦
∆ is an open subset of X.
From [DNNP, Proposition 2.1], using the injectivity of φ and φn,m , it follows
◦
that there is an integer N > n such that for any open set ∆ — the interior of
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
443
a top simplex ∆ ⊂ (X, σ), one has
◦
SP (φn,m ◦ φ)y ∩ ∆ 6= ∅.
We can define the homomorphism ψ : C → Am for each block Ajm of Am
separately. That is, we need to define ψ j : C → Ajm , then let ψ := ⊕ψ j .
If SP(Ajm ) = Xm,j = {pt}, then the partial map (φn,m ◦ φ)j has finite dimensional image, and we can define it to be ψ j . Hence we assume that the
connected finite simplicial complex Xm,j is not the space of single point {pt}.
Let α = (φn,m ◦ φ)j : C → Ajm .
◦
Let Y be the union of all such top simplices ∆ that ∆ ∩ SPα is uncountable.
Let Z be the union of all simplices ∆ which are not top simplices. Both Y and
Z are closed subset of X. Let ∆1 , ∆2 , · · · , ∆l be the list of all top simplices
such that ∆i 6⊂ Y, i = 1, 2, · · · , l. Then
X = Y ∪ ∆ 1 ∪ ∆2 · · · ∪ ∆ l .
(This fact will be used later.) (Here we use the fact that X is equal to the
union of all top simplices, since each simplex is a face of a top simplex.)
◦
For each ∆i , ∆i ∩ SPα is a countable nonempty set. There is a point xi and
an open disk Ui = Bεi (xi ) ∋ xi such that
SPα ∩ Ui = {xi }.
◦
We can assume that Ui ⊂ ∆i . Obviously, ∂∆i is a deformation retract of
∆i \U
¡ i. ¡
¢¢
Set X\ ∪li=1 Ui ∩ SPα = T . Then SPα = T ∪ {x1 , x2 , · · · , xl }.
Define a function g : T → Y ∪ Z(⊂ X) as below.
Let g ′ : Z → Y ∪ Z be the identity map, that is,
g ′ (z) = z,
∀z ∈ Z.
We will extend the map g ′ to a map (let us still denote it by g ′ ).
¡
¢
g ′ : X\ ∪li=1 Ui −→ Y ∪ Z.
For each top simplex ∆ ⊂ Y , extend g ′ |∂∆ to a map g ′ : ∆ → ∆ satisfying
g ′ (T ∩ ∆) = ∆.
◦
(Such extension exists since T ∩ ∆ is uncountable, see Lemma 2.6 of [EGL].)
For any simplex ∆i , i = 1, 2, · · · , l, one can extend g ′ |∂∆ to a map g ′ : ∆i \Ui →
∂∆i , since ∂∆i is a deformation retract of ∆i \Ui .
Documenta Mathematica 7 (2002) 255–461
444
Guihua Gong
¡
¢
Thus we obtain the extension g ′ : X\ ∪li=1 Ui −→ Y ∪ Z. Let g = g ′ |T . Then
g(T ) ⊃ Y,
and
dist(g(x), x) < η, ∀x ∈ T.
Since SPα = T ∪ {x1 , x2 , · · · , xl }, there are homomorphisms α0 : C(T ) → Ajm
| = C({xi }) → Aj , j = 1, 2, · · · , l, with mutually orthogonal images,
and αi : C
m
such that
l
X
(f |{xi } ),
∀f ∈ C(X).
α(f ) = α0 (f |T ) +
i=1
Define β0 : C(Y ∪ Z) →
Ajm
by
β0 (f ) = α0 (f ◦ g),
∀f ∈ C(Y ∪ Z),
where g : T → Y ∪ Z is defined as above. For each ∆i , there is a surjective
map gi : Xm,j → ∆i , since Xm,j 6= {pt}. Define βi : C(∆i ) → Ajm by
βi (f )(x) = f (gi (x)) · αi (1C
| ),
∀f ∈ C(∆i ), x ∈ Xm,j .
Then, obviously, we have
(1′ ) β0 (1C(Y ∪Z) ) = α0 (1C(T ) ), and βi (1C(∆i ) ) = αi (1C
| ), for i = 1, 2, · · · , l.
From the way η is chosen and the properties that dist(g(x), x) < η for any
x ∈ T , and that diameter(∆i ) < η for any i = 1, 2, · · · , l, we have
(2′ ) kβ0 (f |Y ∪Z ) − α0 (f |T )k < ε, and kβi (f |∆i ) − αi (f |{xi } )k < ε for i =
1, 2, · · · , l, and f ∈ F .
Finally, let the partial homomorphism ψ j : C(X) → Ajm be defined by
ψ j (f ) = β0 (f |Y ∪Z ) +
l
X
i=1
βi (f |∆i ).
Since T ⊂ SPα0 and the map g : T → Y ∪ Z satisfies g(T ) ⊃ Y , we have
SP(β0 ) ⊃ Y . Hence SPψ j = SPβ0 ∪ ∪li=1 SPβi ⊃ Y ∪ ∪li=1 ∆i = X. That is, ψ j
is injective.
The property (1) follows from (1′ ) and (2) follows from (2′ ).
⊔
⊓
We will use 5.32a, 5.32b, 1.6.9, 1.6.29, 1.6.30 to prove the following main theorem of this article.
Ltn
M[n,i] (C(Xn,i )), φn,m ) is a simple
Theorem 6.3. Suppose that lim(An = i=1
inductive limit C ∗ -algebra with dim(Xn,i ) ≤ M for a fixed positive integer M .
Ltn
Then there is another inductive system (Bn =
i=1 M{n,i} (C(Yn,i )), φn,m )
with the same limit algebra as the above system, where all Yn,i are spaces of
forms {pt}, [0, 1], S 1 , S 2 , TII,k , or TIII,k .
Proof: Without loss of generality, assume that the spaces Xn,i are connected
finite simplicial complexes and the connecting maps φn,m are injective (see
Theorem 4.23).
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
445
Let
P ε1 > ε2 > ε3 > · · · > 0 be a sequence of positive numbers satisfying
εn < +∞.
We need to construct the intertwining diagram
F1
T
Ax
s(1)
α1
B
S1
E1
F2
φs(1),s(2)
−→
β1
ց
ψ1,2
−→
T
Ax
s(2)
α2
B
S2
Fn+1
Fn
φs(2),s(3)
−→
T
· · · −→ Axs(n)
αn
β2
ց
ψ2,3
−→
· · · −→ B
Sn
E2
φs(n),s(n+1)
−→
βn
ց
ψn,n+1
−→
T
Ax
s(n+1)
αn+1
BS
n+1
En
−→ · · ·
ց
−→ · · ·
En+1
satisfying the following conditions.
(0.1) (As(n) , φs(n),s(m) ) is a sub-inductive system of (An , φn,m ). (Bn , ψn,m )
is an inductive system of matrix algebras over the spaces: {pt}, [0, 1], S 1 ,
∞
2
{TII,k }∞
i=2 , {TIII,k }i=2 , S .
∞
(0.2) Choose {aij }j=1 ⊂ As(i) and {bij }∞
j=1 ⊂ Bi to be countable dense subsets
of the unit balls of As(i) and Bi , respectively. Fn are subsets of the unit balls
of As(n) , and En are subsets of the unit balls of Bn satisfying
φs(n),s(n+1) (Fn ) ∪ αn+1 (En+1 ) ∪
n+1
[
i=1
φs(i),s(n+1) ({ai1 , ai2 , · · · , ai
n+1 })
⊂ Fn+1
and
ψn,n+1 (En ) ∪ βn (Fn ) ∪
n+1
[
i=1
ψi,n+1 ({bi1 , bi2 , · · · , bi
n+1 })
⊂ En+1 .
(Here we use the convention that φn,n = id : An → An .)
(0.3) βn are Fn − 2εn multiplicative and αn are homomorphisms.
(0.4) kψn,n+1 (g) − βn ◦ αn (g)k < 2εn
for all g ∈ En ,
kφs(n),s(n+1) (f ) − αn+1 ◦ βn (f )k < 12εn
for all f ∈ Fn .
and
(0.5) For any block Bni of Bn and any block Ajs(n) of As(n) , the map αni,j satisfies
the following dichotomy condition:
either αni,j is injective or αni,j has a finite dimensional image.
The diagram will be constructed inductively.
First, let B1 = {0}, As(1) = A1 , α1 = 0. Let b1j = 0 ∈ B1 for j = 1, 2, · · · ,
and let {a1j }∞
j=1 be a countable dense subset of the unit ball of As(1) . And let
E1 = {b11 } = B1 and F1 = {a11 } ⊂ As(1) .
Documenta Mathematica 7 (2002) 255–461
446
Guihua Gong
As an inductive assumption, assume that we already have the diagram
F
T1
φs(1),s(2)
−→
As(1)
x
α1
β1
ց
ψ1,2
B
S1
−→
Fn ⊂
Axs(n)
αn
E1
F
T2
As(2)
x
α2
φs(2),s(3)
−→
β2
ց
ψ2,3
−→
B
S2
···
···
···
F
Tn
−→ As(n)
x
ցβn−1 αn
−→
B
Sn
E2
En
and, for each i = 1, 2 · · · , n, we have countable dense subsets {aij }∞
j=1 ⊂
unit ball of As(i) and {bij }∞
⊂
unit
ball
of
B
to
satisfy
the
conditions
(0.1)i
j=1
(0.5) above. We have to construct the next piece of the diagram,
En ⊂
Bn
φs(n),s(n+1)
−→
βn
ց
ψn,n+1
−→
⊃ Fn+1
As(n+1)
x
αn+1
⊃ En+1
Bn+1
,
to satisfy the conditions (0.1)-(0.5).
Our construction are divided into several steps. In order to provide the reader
with a whole picture of the construction, we first give an outline of it. Then
the detailed construction will follow.
Outline of the construction. We will construct the following diagram.
C
✄
✄
ξ1
✄
A✄
✄
✄
s(n)
ξ2
✗✄
✄
✲ P1 Am1 P1
≈φs(n),m1 L
θ
n
RAm2 R
¡
✒
φm1 ,m2
¡
✞
¡
u ✆
✝
✻
✲ P0 Am1 P0 ¡
✒
¡
✻✞
¡
α
✝1 ✆
¡ ✞
✻
✝2
¡
ψ
¡
B
αn
φm ,s(n+1)
1
✲ φm1,s(n+1)(P1)As(n+1)φm1,s(n+1)(P1)
✲D
ξ3
◗
◗
◗
θ0+θ1
◗
ξ4
L
✲ (Q2)As(n+1) (Q2)
≈φm2,s(n+1)
L
◗
s
✲ (Q0 +Q1)As(n+1) (Q0 +Q1)
✒
¡
✲
✄ ¡
λ◦α′ ¡
✞
✿
β
Adu
✘✘✘
¡
✘
✝3 ✆
✡✠
✻
✘✘
✘
¡
✆
′′
✘
✻ ❄
✘ α
¡✘✘✘
✲ B ✘
RAm2 R
θ0+θ1
This large picture consists of several smaller diagrams, each of which is called
a sub-diagram. There are two kinds of sub-diagrams. The sub-diagrams of
the first kind are labeled by the numbers 1, 2, 3 and the letter u (in the
centers of the sub-diagrams). These sub-diagrams are almost commutative in
some sense. For example, the one in the center of the large picture, labeled
by the letter u consists of two composite maps (θ0 + θ1 ) ◦ (λ ◦ α′ ) ◦ β and
(θ0 + θ1 ) ◦ (φm1 ,m2 |P0 Am1 P0 ). They are almost equal to each other on a given
finite set up to unitary equivalence.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
447
The sub-diagrams of the second kind are those two labeled by “≈ φs(n),m1 ” and
“≈ φm2 ,s(n+1) ”. They describe the approximate decompositions of the given
maps “φs(n),m1 ” and “φm2 ,s(n+1) |RAm2 R ”.
All the maps in the above picture are homomorphisms except β, θ, and θ 0 +
θ1 (which are represented by broken line arrows). These maps are linear ∗contractions which are almost multiplicative on some given finite sets (i.e., on
the sets Fn ⊂ As(n) , F := θ(Fn ) ⊂ P0 Am1 P0 , or a certain (large enough) finite
subset F ′ ⊂ RAm2 R2 ) to within given small numbers (i.e., εn or some related
small numbers).
The sub-diagrams labeled by the numbers 1, 2, and 3 are approximately commutative on certain given finite sets (i.e., En ⊂ Bn , G := ψ(En ) ∪ β(F ) ⊂ B
(F is from the above paragraph)) to within a small number (i.e., εn ). The subdiagram labeled by the letter u is approximately commutative on a finite set
(F := θ(Fn )) to within a given small number (9εn ) up to unitary equivalence.
The sub-diagrams labeled by “≈ φs(n),m1 ” and “≈ φm2 ,s(n+1) ” are approximate
decompositions of φs(n),m1 and φm2 ,s(n+1) |RAm2 R , respectively. (E.g., the direct
sum θ ⊕ (ξ2 ◦ ξ1 ) of the two maps θ and ξ2 ◦ ξ1 is close to φs(n),m1 to within a
small number εn on a given finite set Fn .)
The above decomposition of φs(n),m1 and the almost commutative sub-diagram
labeled by the number 1 are obtained in Step 1 in the detailed proof, applying
Theorem 5.32b to As(n) and αn : Bn → As(n) (and to the finite sets En
and Fn ). The main purpose of this step is to make the set θ(Fn ) := F weakly
approximately constant to within εn (the other part ξ2 ◦ξ1 of the decomposition
factors through an interval algebra C), which will be useful later when we apply
Theorem 1.6.9. (If one assumes in the beginning that the set Fn is weakly
approximately constant to within εn , then he does not need this step.)
The sub-diagrams labeled by 2 or u will be explained by another picture later.
The almost commutative sub-diagram labeled by the number 3 and the decomposition of φm2 ,s(n+1) |RAm2 R (i.e., “≈ φm2 ,s(n+1) ” in the picture), are obtained
in Step 4, applying Theorem 5.32a to RAm2 R and λ ◦ α′ : B → RAm2 R (and
certain finite subsets of B and RAm2 R). The purpose of applying Theorem
5.32a is to construct the map θ0 + θ1 to satisfy the condition in Theorem 1.6.9
for the two homotopic homomorphisms λ ◦ (φ ⊕ r) and φm1 ,m2 |P0 Am1 P0 in the
next picture, and therefore to obtain the almost commutative sub-diagram up
to unitary equivalence—the sub-diagram labeled by u—, (the other part ξ4 ◦ ξ3
of the decomposition factors through an interval algebra D).
In order to get the parts of the sub-diagram labeled by 2 and u, we need to
start with α : Bn → P0 Am1 P0 . We describe it in the next picture.
Documenta Mathematica 7 (2002) 255–461
448
Guihua Gong
RAm2 R
✻
λ
θ0+θ1
φm1 ,m2 ✏✏
✶
✏✏ ✞h
✏
✏
✝ ✆
✻
φ ⊕ r✲
ML (A) ⊕ r(A)
A(= P0 Am1 P0 )
✞
◗
4
✞
✻
◗ ✝ ✆
✻′✻
✝2 ✆
α
α
✻ β◗
◗
ψ
s B
✲◗
Bn
✲ (Q0 +Q1)As(n+1) (Q0 +Q1)
✲
✄ ☎
Adu
✍✌
By Corollary 1.6.29 (see 1.6.31 also), applied to the homomorphism α from the
first picture, we obtain the almost commutative sub-diagrams labeled by the
numbers 2 and 4. Then we apply Lemma 1.6.30 to obtain the sub-diagram
labeled by the letter h which commutes up to homotopy equivalence. By Theorem 1.6.9 and the property of the map θ0 + θ1 (from Theorem 5.32a), this
sub-diagram leads to the sub-diagram labeled by u in the first picture.
With the first picture in mind, we define
Bn+1 = C ⊕ B ⊕ D,
ψn,n+1 = (ξ1 ◦ αn ) ⊕ ψ ⊕ (ξ3 ◦ φm1 ,m2 |P0 Am1 P0 ◦ α),
βn = ξ1 ⊕ (β ◦ θ) ⊕ (ξ3 ◦ φm1 ,m2 |P0 Am1 P0 ◦ θ),
and
αn+1 = (φm1 ,s(n+1) |P1 Am1 P1 ◦ ξ2 ) ⊕ (Adu ◦ α′′ ) ⊕ ξ4 .
In the definitions of ψn,n+1 and αn+1 , we use solid line arrows only since these
maps are supposed to be homomorphisms (but in the definition of βn , we can
use broken line arrows).
One can easily verify the conditions (0.1)–(0.5) except that the map
φm1 ,s(n+1) |P1 Am1 P1 ◦ ξ2 may not automatically satisfy the dichotomy condition
(0.5), for which we have to apply Lemma 6.2 to make some modification.
Details of the construction. The above outline can be used as a guide
to understand the following construction. But the proof below is complete by
itself. (We encourage readers to compare the following detailed proof with the
two diagrams in the outline.)
Among the conditions in the induction assumption, only the dichotomy condition (0.5) of αn is used in the following construction.
Step 1. By Theorem 5.32b, applied to αn : Bn → As(n) , En ⊂ Bn , Fn ⊂ As(n) ,
and ε > 0, there are Am1 (m1 > s(n)), two orthogonal projections P0 , P1 ∈ Am1
with φs(n),m1 (1As(n) ) = P0 + P1 and P0 trivial, a C ∗ -algebra C — a direct sum
| —, a unital map θ ∈ Map(A
of matrix algebras over C[0, 1] or C
s(n) , P0 Am1 P0 )1 ,
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
449
a unital homomorphism ξ1 ∈ Hom(As(n) , C)1 , an injective unital homomorphism ξ2 ∈ Hom(C, P1 Am1 P1 )1 and a (not necessarily unital) homomorphism α ∈ Hom(Bn , P0 Am1 P0 ) such that
for all f ∈ Fn .
(1.1) kφs(n),m1 (f ) − θ(f ) ⊕ (ξ2 ◦ ξ1 )(f )k < εn
(1.2) θ is Fn -εn multiplicative and F := θ(Fn ) is weakly approximately constant
to within εn .
(1.3) kα(g) − θ ◦ αn (g)k < εn for all g ∈ En .
(1.4) Both α : Bn → P0 Am1 P0 and ξ2 : C → P1 Am1 P1 satisfy the dichotomy
condition in (0.5).
(Thus we finished the construction of the sub-diagrams labeled by the number
“1” and “≈ φs(n),m1 ” of the large diagram in the outline.)
Let all the blocks of C be parts of C ∗ -algebra Bn+1 . That is,
Bn+1 = C ⊕ (some other blocks).
The map βn : As(n) → Bn+1 and the homomorphism ψn,n+1 : Bn → Bn+1 are
defined by
βn = ξ1 : As(n) → C(⊂ Bn+1 ) and ψn,n+1 = ξ1 ◦ αn : Bn → C ( ⊂ Bn+1 )
for the blocks of C ( ⊂ Bn+1 ). For this part, βn is also a homomorphism.
Step 2. Let A = P0 Am1 P0 , F = θ(Fn ). Since P0 is a trivial projection,
M
Mli (C(Xm1 ,i )).
A∼
=
L
| ) ⊂ A, and r : A → rA be the homomorphism defined by
Let rA :=
Mli (C
evaluation at certain base points x0i ∈ Xm1 ,i (see 1.1.7(h)).
Applying Corollary 1.6.29 (see Remark 1.6.31 also) to α : Bn → A (notice
that α satisfies the dichotomy condition), En ⊂ Bn and F ⊂ A, we obtain the
following diagram:
φ⊕r
A −→ ML (A)x⊕ r(A)
x
′
ցβ
α
α
Bn
ψ
−→
B
such that
(2.1) B is a direct sum of matrix algebras over {pt}, [0, 1], S 1 , TII,k , TIII,k , or
S2.
(2.2) α′ is an injective homomorphism, and β is an F -εn multiplicative map.
(2.3) φ : A → ML (A) is a unital simple embedding. r : A → r(A) is the
homomorphism defined by evaluations as in 1.1.7(h).
(2.4) kβ ◦ α(g) − ψ(g)k < εn for all g ∈ En ,
k(φ ⊕ r)(f ) − α′ ◦ β(f )k < εn
for all f ∈ F (:= θ(Fn )).
(Thus we finished the construction of the sub-diagrams labeled by the number
“2” and “4” of the second diagram in the outline.)
Let all the blocks B be also parts of Bn+1 , that is,
Bn+1 = C ⊕ B ⊕ (some other blocks).
Documenta Mathematica 7 (2002) 255–461
450
Guihua Gong
The maps βn : As(n) → Bn+1 , ψn,n+1 : Bn → Bn+1 are defined by
θ
β
βn := β ◦ θ : As(n) −→ A −→ B (⊂ Bn+1 )
and
ψn,n+1 := ψ : Bn −→ B (⊂ Bn+1 )
for the blocks of B(⊂ Bn+1 ). This part of βn is Fn -2εn multiplicative, since θ
is Fn -εn multiplicative, β is F -εn multiplicative, and F = θ(Fn ).
Step 3. By the simplicity of lim(An , φn,m ), for m large enough, the homomorphism φm1 ,m |P0 Am1 P0 is 4M -large in the sense of 1.6.16. By Lemma 1.6.30,
applied to φ ⊕ r : A → ML (A) ⊕ r(A), there is an Am2 and unital homomorphism
L j Lλ : jML (A) ⊕ r(A) → RAm2 R, where R = φm1 ,m2 (P0 ) (write R as
jR ∈
j Ami ) such that the diagram
RAm2 R
φm1 ,m2
✶
✏✏
✻
✏
✏✏
λ
✏
✏
φ ⊕ r✲
A(= P0 Am1 P0 )
ML (A) ⊕ r(A)
satisfies the following conditions:
(3.1) For each block Ajm2 , the partial map
λ·,j : ML (A) ⊕ r(A) −→ Rj Ajm2 Rj
is non zero. Furthermore, either it is injective or it has finite dimensional image
— depending on whether SP(Ajm2 ) is a single point space.
(3.2) λ ◦ (φ ⊕ r) is homotopy equivalent to
φ′ := φm1 ,m2 |A .
(Thus we finished the construction of the sub-diagram labeled by the letter “h”
of the second diagram in the outline.)
Step 4. Applying Theorem 1.6.9 to the finite set F ⊂ A (which is weakly
approximately constant to within εn ), and to two homotopic homomorphisms
φ′ and λ ◦ (φ ⊕ r) : A −→ RAm2 R
(with RAm2 R in place of C), we obtain a finite set F ′ ⊂ RAm2 R, δ > 0 and
L > 0 as in the Theorem 1.6.9.
Let G := ψ(En ) ∪ β(F ) ⊂ B. By Theorem 5.32a, applied to RAm2 R,
λ ◦ α′ : B −→ RAm2 R
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
451
(which satisfies the dichotomy condition by (2.2) and (3.1)), finite sets G ⊂
B, F ′ ⊂ RAm2 R, min(εn , δ) > 0 (in place of ε), and L > 0, there are As(n+1) ,
mutually orthogonal projections Q0 , Q1 , Q2 ∈ As(n+1) with φm2 ,s(n+1) (R) =
Q0 + Q1 + Q2 , a C ∗ -algebra D — a direct sum of matrix algebras over
C[0, 1]—, a unital map θ0 ∈ Map(RAm2 R, Q0 As(n+1) Q0 ) and four unital homomorphisms θ1 ∈ Hom(RAm2 R, Q1 As(n+1) Q1 )1 , ξ3 ∈ Hom(RAm2 R, D)1 , ξ4 ∈
Hom(D, Q2 As(n+1) Q2 )1 , and α′′ ∈ Hom(B, (Q0 + Q1 )As(n+1) (Q0 + Q1 ))1 such
that the following are true:
(4.1) kφm2 ,s(n+1) (f ) − ((θ0 + θ1 ) ⊕ (ξ4 ◦ ξ3 ))(f )k < εn for all f ∈ F ′ ⊂ RAm2 R.
(4.2) kα′′ (g) − (θ0 + θ1 ) ◦ λ ◦ α′ (g)k < εn for all g ∈ G.
(4.3) θ0 is F ′ -min(εn , δ) multiplicative and θ1 satisfies that
θ1i,j ([q]) > L · [θ0i,j (Ri )]
for any non zero projection q ∈ Ri Am2 Ri .
(4.4) Both α′′ : B −→ (Q0 + Q1 )As(n+1) (Q0 + Q1 ) and ξ4 : D → Q2 As(n+1) Q2
satisfy the dichotomy condition (0.5).
(Thus we finished the construction of the sub-diagrams labeled by the number
“3” and “≈ φm2 ,s(n+1) ” of the large diagram in the outline. Combined with
Step 2 and Step 3, these two sub-diagrams will lead to the sub-diagram labeled
by the letter “u” of the large diagram as below.)
By the end of 1.1.4, for any blocks Ai , Akm2 and any non zero projection e ∈
k
Ai , φi,k
m1 ,m2 (e) ∈ Am2 is a non zero projection. As a consequence of (4.3), we
have
[(θ1 ◦ φ′ )(e)] ≥ L · [θ0 (R)] (= L · [Q0 ]),
(Recall that φ′ = φm1 ,m2 |A ). Therefore, θ0 and θ1 (in place of λ0 and λ1 )
satisfy the condition in Theorem 1.6.9. By Theorem 1.6.9, there is a unitary
u ∈ (Q0 + Q1 )As(n+1) (Q0 + Q1 ) such that
k(θ0 + θ1 ) ◦ φ′ (f ) − Adu ◦ (θ0 + θ1 ) ◦ λ ◦ (φ ⊕ r)(f )k < 8εn ,
∀f ∈ F.
Combining it with the second inequality of (2.4), we have
(4.5) k(θ0 + θ1 ) ◦ φ′ (f ) − Adu ◦ (θ0 + θ1 ) ◦ λ ◦ α′ ◦ β(f )k < 9εn ,
∀f ∈ F.
Step 5. Finally, let all the blocks of D be the rest of Bn+1 . Namely, let
Bn+1 = C ⊕ B ⊕ D,
where C is from Step 1, B is from Step 2, and D is from Step 4.
We already have the definitions of βn : As(n) → Bn+1 and ψn,n+1 : Bn → Bn+1
for those blocks of C ⊕ B ⊂ Bn+1 (from Step 1 and Step 2). The definitions of
βn and ψn,n+1 for blocks of D, and the homomorphism αn+1 : C ⊕ B ⊕ D →
As(n+1) will be given below.
The part of βn : As(n) → D (⊂ Bn+1 ) is defined by
θ
φ′
ξ3
βn = ξ3 ◦ φ′ ◦ θ : As(n) −→ A −→ RAm2 R −→ D.
Documenta Mathematica 7 (2002) 255–461
452
Guihua Gong
(Recall that A = P0 Am1 P0 and φ′ = φm1 ,m2 |A .) Since θ is Fn -εn multiplicative,
and φ′ and ξ3 are homomorphisms, we know that this part of βn is Fn -εn
multiplicative.
The part of ψn,n+1 : Bn → D (⊂ Bn+1 ) is defined by
φ′
α
ξ3
ψn,n+1 = ξ3 ◦ φ′ ◦ α : Bn −→ A −→ RAm2 R −→ D
which is a homomorphism. The homomorphism αn+1 : C ⊕ B ⊕ D → As(n+1)
is defined as follows.
Consider the composition
φ′′
ξ2
φ′′ ◦ ξ2 : C −→ P1 Am1 P1 −→ φm1 ,s(n+1) (P1 )As(n+1) φm1 ,s(n+1) (P1 ),
where P1 and ξ2 are from Step 1, φ′′ = φm1 ,s(n+1) |P1 Am1 P1 . Using the dichotomy condition of ξ2 , by Lemma 6.2, there is a homomorphism τ : C →
φm1 ,s(n+1) (P1 )As(n+1) φm1 ,s(n+1) (P1 ) such that
(5.1)
kτ (f ) − (φ′′ ◦ ξ2 )(f )k < εn , ∀f ∈ ξ1 (Fn ) ⊂ C, and
(5.2) τ satisfies the dichotomy condition (0.5).
Define
αn+1 |C = τ : C → φm1 ,s(n+1) (P1 )As(n+1) φm1 ,s(n+1) (P1 ),
✞ ☞
α′′
αn+1 |B = Adu ◦ α′′ : B −→ (Q0 + Q1 )As(n+1) (Q0 + Q1 )
Adu ,
✝
✻ ✌
where α′′ is from Step 4, and define
αn+1 |D = ξ4 : D −→ Q2 As(n+1) Q2 .
∞
Finally, choose {an+1 j }∞
j=1 ⊂ As(n+1) and {bn+1 j }j=1 ⊂ Bn+1 to be countable
dense subsets of unit balls of As(n+1) and Bn+1 , respectively. And choose
Fn+1 = φs(n),s(n+1) (Fn ) ∪ αn+1 (En+1 ) ∪
n+1
[
i=1
φs(i),s(n+1) ({ai1 , ai2 , · · · , ai
n+1 })
and
En+1 = ψn,n+1 (En ) ∪ βn (Fn ) ∪
n+1
[
i=1
ψi,n+1 ({bi1 , bi2 , · · · , bi
n+1 }).
Thus we obtain the following diagram:
Fn ⊂
En ⊂
Axs(n)
αn
Bn
φs(n),s(n+1)
−→
βn
ց
ψn,n+1
−→
As(n+1)
x
αn+1
Bn+1
⊃ Fn+1
⊃ En+1 .
Step 6. Now we need to verify all the conditions (0.1)–(0.5) for the above
diagram.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
453
(0.1)–(0.2) hold from the construction (see the constructions of B, C, D in Step
1, Step 2 and Step 4, and En+1 , Fn+1 in the end of Step 5.)
(0.3) follows from the end of Step 1, the end of Step 2, and the part of the
definition of βn for D from Step 5.
(0.5) follows from (4.4) and (5.2).
So we only need to verify (0.4).
Combining (1.1) with (4.1), we have
kφs(n),s(n+1) (f ) − [(φ′′ ◦ ξ2 ◦ ξ1 ) ⊕ ((θ0 + θ1 ) ◦ φ′ ◦ θ) ⊕ (ξ4 ◦ ξ3 ◦ φ′ ◦ θ)](f )k
< εn + εn = 2εn
for all f ∈ Fn (recall that φ′′ = φm1 ,s(n+1) |P1 Am1 P1 , φ′ := φm1 ,m2 |P0 Am1 P0 ).
Combined with (4.2), (4.5), (5.1) and the definitions of βn and αn+1 , the proceeding inequality yields
kφs(n),s(n+1) (f ) − (αn+1 ◦ βn )(f )k < 9εn + εn + 2εn = 12εn
∀f ∈ Fn .
Combining (1.3), the first inequality of (2.4), and the definitions of βn and
ψn,n+1 , we have
kψn,n+1 (g) − βn ◦ αn (g)k < εn + εn = 2εn
∀g ∈ En .
So we obtain (0.4).
The theorem follows from Proposition 6.1.
⊔
⊓
Remark 6.4. In the proof of the above theorem, if there is at least one block
of Bn+1 having spectrum of forms S 1 , TII,k , TIII,k , or S 2 , then we can chose
the map ψn,n+1 to be injective (e.g., the map ψ in Step 2 can be chosen to
be injective). Hence, in general, we can make the maps ψn,m , in the inductive
system (Bn , ψn,m ), injective. (Note that if no space of S 1 , TII,k , TIII,k or S 2
appears, then it is easy to make the maps injective; see Theorem 2.2.1 of [Li2]).
Remark 6.5. By Lemma 1.3.3, our main result Theorem 6.3 also holds for
∗
general simple
Ltn AH inductive limit C -algebras
lim(An = i=1 Pn,i M[n,i] (C(Xn,i ))Pn,i , φn,m ) with uniformly bounded dimensions of Xn,i , where Pn,i ∈ M[n,i] (C(Xn,i )) are projections. That is, such
an AH algebra can be written as an inductive limit of a system (Bn =
L
sn
i=1 Qn,i M{n,i} (C(Yn,i ))Qn,i , ψn,m ), where Yn,i are the spaces:
{pt}, [0, 1], S 1 , TII,k , TIII,k and S 2 , and Qn,i ∈ M{n,i} (C(Yn,i )) are projections.
6.6.
Suppose that a simple C ∗ -algebra A is an inductive limit of matrix
algebras over Xn,i , where Xn,i are the spaces of forms {pt}, [0, 1], S 1 , S 2 , TII,k
or TIII,k . Suppose that K∗ (A) is torsion free. Then it can be proved that for
each fixed algebra An , integer N > 0, there is an Am such that
rankφi,j
n,m (1Ain )
≥ N,
rank(1Ain )
Documenta Mathematica 7 (2002) 255–461
454
Guihua Gong
and that (φn,m )∗ (torK∗ (An )) = 0. Based on this, using the argument
from §4 of [G2], we know that for any F ⊂ An , ε > 0, if N is large
enough, then the above φn,m is homotopic to a homomorphism ψ : An →
φn,m (1An )Am φn,m (1An ) satisfying ψ(F ) ⊂ε C, where C is a direct sum of matrix algebras over spaces {pt}, [0, 1] and S 1 . (See [G1] and the proof of Lemma
5.6 of [EGL] also.) Using the above fact, the following Corollary is a direct
consequence of our Main Theorem and its proof. (In fact, since the algebras
| ), Mk (C[0, 1]) and Mk (C(S 1 )) are stably generated, the proof is much
Mk (C
simpler (see §3 of [Li3]).
Corollary 6.7. Suppose that A is a simple C ∗ -algebra which is an inductive
limit of an AH system with uniformly bounded dimensions of local spectra.
If K∗ (A) is torsion free, then it is an inductive limit of matrix algebras over
C(S 1 ).
Combining the above corollary with [El2] (see [NT] also), we have the following
theorem.
Ltn
Pn,i M[n,i](C(Xn,i ))Pn,i ,φn,m )
Theorem 6.8. Suppose that A= lim (An = i=1
n→∞
Ls n
and B = lim (Bn =
i=1 Qn,i M{n,i} (C(Yn,i ))Qn,i , ψn,m ) are unital simple
n→∞
inductive limit algebras with uniformly bounded dimensions of local spectra X n,i
and Yn,i , respectively. Suppose that K∗ (A) = K∗ (B) are torsion free.
Suppose that there is an isomorphism of ordered groups
φ0 : K0 A −→ K0 B
taking [1] ∈ K0 A into [1] ∈ K0 B, that there is a group isomorphism
φ1 : K1 A −→ K1 B
and that there is an isomorphism between compact convex sets
φT : T B −→ T A,
where T A and T B denote the simplices of tracial states of A and B, respectively.
Suppose that φ0 and φT are compatible, in the sense that
τ (φ0 g) = φT (τ )(g),
g ∈ K0 A, τ ∈ T B.
It follows that there exists an isomorphism
φ : A −→ B
giving rise to φ0 , φ1 , φT .
Remark 6.9. Since the C ∗ -algebras C(TII,k ), C(TIII,k ) and C(S 2 ) are not
stably generated, our proof heavily depends on the results that, certain G-δ
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
455
multiplicative maps (with parts of point evaluations of sufficiently large sizes)
are approximated by true homomorphisms in §5. We believe that such results
should play important role in the future study of general simple C ∗ -algebras
(with or without real rank zero property).
Remark 6.10.
From a result of J. Villadsen, [V1], one knows that the
restriction on the dimensions of the spaces Xn,i can not be removed.
In [G5]—an appendix to this article, we will show that the condition of uniformly bounded dimensions of local spectra can be replaced by the condition
of very slow dimension growth. The main difficulty for this case is that we can
not obtain the homomorphism from Bn to As(n) as the homomorphism αn in
the above proof. (The αn in this case will be only a sufficiently multiplicative
map.) But we can still construct homomorphisms ψn : Bn → Bn+1 , if we
carefully choose αn and βn . This case does not create essential difficulty, but
makes the proof much longer. We refer it to [G5] , a separate appendix to this
paper.
It could be an improvement if one can replace the very slow dimension growth
condition by the slow dimension growth condition. The author believes that
the theorem is also true for this case. In fact, if one can prove the corresponding
decomposition results (see Section 4) for the AH-algebras with slow dimension
growth, then the Main Theorem in this article would also hold, by the same
proof as in [G5].
Ltn
Corollary 6.11. Suppose that A = lim (An = i=1
M[n,i] (C(Xn,i )), φn,m )
n→∞
∗
is a simple inductive limit C -algebras. Suppose that each of the spaces Xn,i
is of the forms: {pt}, [0, 1], S 1 , S 2 , TII,k or TIII,k . And suppose that all the
connecting maps φn,m are injective. For any F ⊂ An , ε > 0, if m is large
enough, then there are two mutually orthogonal projections P, Q ∈ Am and two
homomorphisms φ : An → P Am P and ψ : An → QAm Q such that
(1) kφn,m (f ) − (φ ⊕ ψ)(f )k < ε for all f ∈ F ;
(2) φ(F ) is weakly approximately constant to within ε and SPV(φ) < ε;
(3) ψ factors through matrix algebras over C[0, 1].
i
j
Furthermore, if for some i, j, the partial map φi,j
n,m : An → Am is homotopic
to a homomorphism with finite dimensional image, then the part φ of the
decomposition φ ⊕ ψ corresponding to this partial map can be chosen to be zero
(or, equivalently, φi,j
n,m itself is close to a homomorphism factoring through a
matrix algebra over C[0, 1]).
Proof: It follows from the corollary of 2.3 of [Su] that for any Ml (C(X)),
ε > 0, there are ε1 > 0 and a finite subset F of self adjoint elements
of Ml (C(X)) (i.e., F ⊂ (Ml (C(X)))s.a ) such that for any homomorphism
φ : Ml (C(X)) → Ml1 (C(Y )), if φ(F ) is weakly approximately constant to
within ε1 , then SPV(φ) < ε. Therefore, for the desired condition (2) above, we
only need to make φ(F ) weakly approximately constant to within min(ε, ε 1 ).
Documenta Mathematica 7 (2002) 255–461
456
Guihua Gong
To simplify the notation, we still denote min(ε, ε1 ) by ε.
Now, the main body of the corollary follows from Lemma 6.2 and Theorem
5.32b. Namely, first apply Lemma 6.2 to id : An → An (in place of φ) and An
in place of both B and An to find An1 (in place of Am ) and homomorphism
α : An → An1 such that α satisfies the dichotomy condition and such that α
is sufficiently close to φn,n1 on the finite set F . Then apply Lemma 5.32b to
An and F ⊂ An (in place of B and G ⊂ B), An1 and φn,n1 (F ) ⊂ An1 (in place
of An and F ⊂ A), and α : An → An1 (in place of α : B → An ) to construct
the desired decomposition. (Note that we use the following trivial fact: If two
maps φ1 , φ2 : An → Am are approximately equal to each other to within ε1 on
the finite set F and the set φ1 (F ) is weakly approximately constant to within
ε2 , then the set φ2 (F ) is weakly approximately constant to within 2ε1 + ε2 .)
For the last part of the Corollary, one needs to notice the following facts.
(i) In the additional parts of Corollaries 5.22 and 5.23, if the homomorphisms ψ
are homomorphisms with finite dimensional images, then the homomorphisms φ̃
in the corollaries 5.22 and 5.23 are also homomorphisms with finite dimensional
images.
(ii) In Lemma 5.28, if both φ and φ1 are homomorphisms factoring through
interval algebras (this condition implies that they are homotopic to homomorphisms with finite dimensional images), then the homomorphism ψ in Lemma
5.28 (with [ψ]∗ = [φ2 ]∗ ) can be chosen to be a homomorphism with finite
dimensional image.
With the above facts, if φi,j
n,m is homotopic to a homomorphism with finite
dimensional image and if Xn,i 6= S 1 , then the corresponding part of φ in our
corollary could be chosen to be a homomorphism with finite dimensional image,
and therefore it can also factor through matrix algebras over C[0, 1]. So, we
can put it together with the part ψ and hence the part φ disappears from the
decomposition of this partial map. This proves the additional part for the case
Xn,i 6= S 1 .
For the case that Xn,i = S 1 , the additional part of the corollary follows from
the following claim.
Claim: For any unitary u ∈ Ain and any ε > 0, there is an integer N > n such
that if m > N , and if φi,j
n,m (u) is in the path connected component of the unit
j i,j
in the unitary group of φi,j
n,m (1Ain )Am φn,m (1Ain ), then there is a self adjoint
i,j
j i,j
element a ∈ φn,m (1Ain )Am φn,m (1Ain ) such that
2πia
kφi,j
k < ε.
n,m (u) − e
(Obviously, if φi,j
n,m is homotopic to a homomorphism with finite dimensional
image, then φi,j
n,m (u) is in the path connected component of the unit element
j i,j
in the unitary group of φi,j
n,m (1Ain )Am φn,m (1Ain ).)
The proof of the above claim is exactly the same as the proof of the main
theorem of [Phi3]: the simple inductive limit C ∗ -algebra in our corollary has
exponential rank at most 1 + ε. We omit the details.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
457
We point out that, in [EGL], we will only need this result for the case Xm,j =
S 2 . Since dim(S 2 ) ≤ 2, P M• (C(S 2 ))P has exponential rank at most 1 + ε.
Therefore, the claim for the case Xm,j = S 2 (Xn,i = S 1 ) is trivial.
⊔
⊓
By Lemma
1.3.3,
the
above
corollary
also
holds
for
the
case
of
Ltn
Ltn
An = i=1
Pn,i M[n,i] (C(Xn,i ))Pn,i , instead of An = i=1
M[n,i] (C(Xn,i )).
Corollary 6.12. Suppose that A = lim (An =
n→∞
∗
Ltn
i=1
Pn,i M[n,i] (C(Xn,i ))Pn,i ,
φn,m ) is a simple inductive limit C -algebra. Suppose that each of the spaces
Xn,i is of the forms: {pt}, [0, 1], S 1 , S 2 , TII,k or TIII,k . And suppose that all
the connecting maps φn,m are injective. For any F ⊂ An , ε > 0, if m is large
enough, then there are two mutually orthogonal projections P, Q ∈ Am and two
homomorphisms φ : An → P Am P and ψ : An → QAm Q such that
(1) kφn,m (f ) − (φ ⊕ ψ)(f )k < ε for all f ∈ F ;
(2) φ(F ) is weakly approximately constant to within ε and SPV(φ) < ε;
(3) ψ factors through matrix algebras over C[0, 1].
j
i
Furthermore, if for some i, j, the partial map φi,j
n,m : An → Am is homotopic
to a homomorphism with finite dimensional image, then the part φ of the
decomposition φ ⊕ ψ corresponding to this partial map can be chosen to be zero
(or, equivalently, φi,j
n,m itself is close to a homomorphism factoring through a
matrix algebra over C[0, 1]).
Proof: By Lemma 1.3.3, there is an inductive system
à = lim (A˜n =
n→∞
tn
M
M{n,i} (C(Xn,i )), φ̃n,m )
i=1
such that each Pn,i M[n,i] (C(Xn,i ))Pn,i is a corner of M{n,i} (C(Xn,i )) and
φn,m = φ̃n,m |Pn,i M[n,i] (C(Xn,i ))Pn,i . Ã is simple since it is stably isomorphic
to a simple C ∗ -algebra A. φ̃n,m are injective since φn,m are injective. Apply
n
⊂ An ⊂ Ãn and 4ε > 0 to obtain φ̃ and ψ̃ as
Corollary 6.11 to F ∪ {1Ain }ti=1
the homomorphisms φ and ψ in Corollary 6.11. Since
k(φ̃ + ψ̃)(1Ain ) − φ̃n,m (1Ain )k <
there is a unitary u ∈ Ãm such that ku − 1k <
ε
2
ε
, ∀i,
4
and
u((φ̃ + ψ̃)(1Ain ))u∗ = φ̃n,m (1Ain ) = φn,m (1Ain ), ∀i
Finally, let
φ = (Adu ◦ φ̃)|An
and ψ = (Adu ◦ ψ̃)|An
to obtain our corollary.
⊔
⊓
Documenta Mathematica 7 (2002) 255–461
458
Guihua Gong
References
[Bl1]
B. Blackadar, Matricial and ultra-matricial topology, Operator Algebras, Mathematical Physics, and Low Dimensional Topology (R. H.
Herman and B. Tanbay, eds.), A K Peters, Massachusetts, 1993, pp.
11–38.
[Bl2]
B. Blackadar, K-Theory for Operator Algebras, Springer-Verlag, New
York/Berlin/Heidelberg, 1986.
[Bl3]
B. Blackadar, Symmetries of the CAR Algebras, Ann. of Math.,
131(1990), 589–623.
[BDR] B. Blackadar, M. Dadarlat, and M. Rørdam, M. The real rank of inductive limit C*-algebras, Math. Scand. 69 (1991), 211–216.
[Br]
O. Bratteli, Inductive limits of finite dimensional C ∗ -algebras, Trans.
A.M.S., 171(1972), 195–234.
[Ch]
E. Christensen, Near inclusions of C ∗ -algebras, Acta Math. 144(1980)
249-265.
[Con] A. Connes, Noncommutative Geometry, Academic press, New York,
Tokyo, 1995.
[Cu]
J. Cuntz, K-theory for certain C ∗ -algebras, Ann. Math. 113 (1981),
181-197.
[D1]
M. Dadarlat, Approximately unitarily equivalent morphisms and inductive limit C ∗ -algebras, K-theory 9 (1995), 117–137.
[D2]
M. Dadarlat, Reduction to dimension three of local spectra of real rank
zero C ∗ -algebras, J. Reine Angew. Math. 460 (1995), 189–212.
[DG]
M. Dadarlat and G. Gong, A classification result for approximately
homogeneous C ∗ -algebras of real rank zero , Geometric and Functional
Analysis, 7(1997) 646–711.
[DNNP] M. Dadarlat, G. Nagy, A. Nemethi, and C. Pasnicu, Reduction of
topological stable rank in inductive limits of C*-algebras, Pacific J.
Math. 153 (1992), 267–276.
[DN]
M. Dadarlat and Nemethi, Shape theory and (connective) K-theory,
J. Operator Theory 23 (1990), 207–291.
[Da]
K. Davidson, C ∗ -algebras by examples, Fields Institute Monographs,
6, A.M.S. Providence, R.I.
[Ell1] G. A. Elliott, A classification of certain simple C ∗ -algebras, Quantum
and Non-Commutative Analysis (editors, H. Araki et al.), Kluwer, Dordrecht, 1993, pp. 373–385.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
459
[Ell2] G. A. Elliott, A classification of certain simple C ∗ -algebras, II, J.
Ramanujan Math. Soc. 12 (1997), 97–134.
[Ell3] G. A. Elliott, The classification problem for amenable C ∗ -algebras,
Proceedings of the International Congress of Mathematicians, Zürich,
Switzerland, 1994 (editors, S.D. Chattrji), Birkhäuser, Basel, 1995, pp.
922–932.
[Ell4] Elliott, G.A., On the classification of inductive limits of sequences of
semisimple finite dimensional algebras, J. Algebra 38 (1976), 29–44.
[Ell5] Elliott, G.A., On the classification of C*-algebras of real rank zero, J.
Reine Angew. Math. 443 (1993), 179–219.
[EE]
G. A. Elliott and D. E. Evans, The structure of irrational rotation
C ∗ -algebras, Ann. of Math. 138 (1993), 477–501.
[EG1] G. A. Elliott and G. Gong, On inductive limits of matrix algebras over
two-tori, Amer. J. Math. 118 (1996), 263–290.
[EG2] G. A. Elliott and G. Gong, On the classification of C ∗ -algebras of real
rank zero, II, Ann. of Math. 144 (1996), 497–610.
[EGJS] G. A. Elliott, G. Gong, X. Jiang, and H. Su, A classification of simple
limits of dimension drop C ∗ -algebras, Fields Institute Communications
13 (1997), 125–143.
[EGL] G. A. Elliott, G. Gong, and L. Li, On the classification of simple
inductive limit C ∗ -algebras, II : The isomorphism theorem, preprint.
[EGLP] G. A. Elliott, G. Gong, H. Lin, and C. Pasnicu, Abelian C ∗ subalgebras of C ∗ -algebras of real rank zero and inductive limit C ∗ algebras, Duke Math. J. 83 (1996), 511–554.
[G1]
G. Gong, Approximation by dimension drop C ∗ -algebras and classification, C. R. Math. Rep. Acad. Sci. Canada 16 (1994), 40–44.
[G2]
G. Gong, Classification of C ∗ -algebras of real rank zero and unsuspended E-equivalence types, J. Funct. Anal. 152(1998), 281–329.
[G3]
G. Gong, On inductive limits of matrix algebras over higher dimensional spaces, Part I, Math. Scand. 80 (1997), 45–60.
[G4]
G. Gong, On inductive limits of matrix algebras over higher dimensional spaces, Part II, Math. Scand. 80(1997), 61–100.
[G5]
G. Gong, Simple inductive limit C ∗ -algebras with very slow dimension
growth: An appendix for “On the classification of simple inductive limit
C ∗ -algebras, I: The reduction theorem”, Preprint.
Documenta Mathematica 7 (2002) 255–461
460
Guihua Gong
[GL1] G. Gong and H. Lin, The exponential rank of inductive limit C ∗ algebras,, Math. Scand. 71 (1992), 301–319.
[GL2] G. Gong and H. Lin, Almost multiplicative morphisms and K-theory,
International J. of Math. 11(2000) 983-1000.
[HV]
P. Halmos and H. Vaughan, Marriage problems, Amer. J. of Math. 72
(1950), 214–215.
[Hu]
D. Husemoller, Fibre Bundles, McGraw–Hill, New York, 1966; reprinted
in Springer-Verlag Graduate Texts in Mathematics.
[JS1]
X. Jiang and H. Su, On a simple unital projectionless C ∗ -algebra,
American J. of Math. 121(1999), 359–413.
[JS2]
X. Jiang and H. Su, A classification of simple inductive limits of splitting interval algebras, J. Funct. Anal., 151(1997), 50–76.
[Kir]
E. Kirchberg, The classification of purely infinite C ∗ -algebras using
Kasparov’s theory, preprint.
[Li1]
L. Li, On the classification of simple C ∗ -algebras: Inductive limits of
matrix algebras over trees, Mem. Amer. Math. Soc., no. 605, vol. 127,
1997.
[Li2]
L. Li, Simple inductive limit C ∗ -algebras: Spectra and approximation
by interval algebras, J. Reine. Angew Math. 507 (1999), 57–79.
[Li3]
L. Li, Classification of simple C ∗ -algebras: Inductive limits of matrix
algebras over 1-dimensional spaces, J. Funct. Anal.192 (2002) 1-51.
[Lo]
T. Loring, Lifting solutions to perturbing problems in C ∗ -algebras, 8,
Fields Institute monograph, Providence, R.I.
[NT]
K. Nielsen, and K. Thomsen, Limit of circle algebras, Exposition Math
14(1996) 17–56.
[Phi1] N. C. Phillips, A classification theorem for nuclear purely infinite simple
C ∗ -algebras, Doc. Math. 5 (2000), 49-114.
[Phi2] N. C. Phillips, How many exponentials?, American J. of Math. 116
(1994) 1513–1543.
[Phi3] N. C. Phillips, Reduction of exponential rank in direct limits of C ∗ algebras, Canad. J. Math. 46 (1994), 818–853.
[R]
M. Rørdam, Classification of inductive limits of Cuntz algebras, J.
Reine Angew. Math. 440 (1993), 175–200.
[Sch]
C. Schochet, Topological methods for C ∗ -algebras IV: mod p homology,
Pacific J. Math. 114 (1984), 447–468.
Documenta Mathematica 7 (2002) 255–461
Simple Inductive Limit C ∗ -Algebras, I
461
[Se]
G. Segal, K-homology theory in K-theory and operator algebras, Lecture
Notes in Mathematics 575, Springer-Verlag, 1977, pp. 113–127.
[St]
J. R. Stallings, Lectures on Polyhedral Topology (notes by G. A. Swarup)
1967, Tata Institute of Fundamental Research, Bombay.
[Su]
H. Su, On the classification of C ∗ -algebras of real rank zero: Inductive
limits of matrix algebras over non-Hausdorff graphs, Mem. Amer. Math.
Soc. no. 547, vol. 114 (1995).
[Th1] K. Thomsen, Limits of certain subhomogeneous C ∗ -algebras, Mem.
Soc. Math. Fr. 71 (1999).
[V1]
J. Villadsen, Simple C ∗ -algebras with perforation, J. Funct. Anal.,
154(1998), 110–116.
[V2]
J. Villadsen, On stable rank of Simple C ∗ -algebras, J. of Amer. Math.
Soc. 12(1999), 1091–1102.
[Wh]
G. Whitehead, Elements of Homotopy theory, Springer-Verlag, 1978.
[Zh]
S. Zhang, A property of purely infinite simple C ∗ -algebras,
Amer. Math. Soc. 109 (1990), 717–720.
Guihua Gong
Department of Mathematics
University of Puerto Rico, Rio Piedras
San Juan, PR 00931-3355, USA
ggong@goliath.cnnet.clu.edu
Documenta Mathematica 7 (2002) 255–461
Proc.
462
Documenta Mathematica 7 (2002)
Keep reading this paper — and 50 million others — with a free Academia account
Used by leading Academics
Alexandros Karam
National Institute of Chemical Physics and Biophysics
Florentin Smarandache
University of New Mexico
Taro Kimura
Université de Bourgogne
Gordon N Fleming
Pennsylvania State University
