Nonstandard methods in geometric functional analysis

. Maximum principle.

xo e V@) for which For every formula e of ZFC there is an element W=x)q(x)n : ne(xs)n.

In particular, V@) contains an object ,9 whichplays the role of the field R inside V@) .

2.8. Besides the above principles, there is an important procedure for passing to V@) from the ordinary von Neumann universe V , where the latter is defined by the recursion schema The element x^ € V@) is known as the standard name of x . We thus have a canonical embedding of V into V@) . Apart from this we have a technique of lowerings and liftings of sets and correspondences.

2.9. Given an element x e V@), its lowering x I is defined by the rule x | :-{t e V@l: [/ e xn : l]. The set x J is cyclic, i.e., closed with respect to mixing of its elements.

2.10. Let F be a correspondence from X to I inside V@) . There exists a correspondence f' | -and it is unique-from X I to I I such that for any subset A of X inside V@) ,we have F(A) J : F I (1 l) .

In particular, a map f : Xn + Y inside V@) defines a function-lowering f I: X-+Y I suchthat f J(x) -f(x")(xeX).

2.ll.Let x €P(V@)). Define Al:-A

and dom(xt)-x, im(xJ) -{1}. The element x f is called the lifting of x. It is easy to see that x 1l is the least cyclic set containing x,i.e., its cyclic hull: x lJ : mix(x) .

2.l2.Let X,Y e P1V@l) andlet F beacorrespondenceform X to Y. There exists a correspondence ^F | -and it is unique-from X I to Y I inside V@) such that dom(,F 1) : dom(f') f and for every subset A of dom(f') we have F I @ I) : F (A) 1 if and only if F is extensional, i.e., !1e F(x,) r llxr : x.rn 3 V W1: !2n.

YeF@r)

Inparticular, amap f:X ---+Y J generatesafunction f I:X" --+ Y such that f t(x") -f(x) for x e X.If necessaryinspecificcases,thelowering and lifting procedures can be iterated. $3. Vector lattices 3.1. There are several excellent monographs on the theory of vector lattices [4], [18], [19], [55], [70]. Vector lattices are also commonly known as Riesz spaces, and order-complete vector lattices as Kantorovich spaces or K-spaces. A K-space is said to be extended if any set of pairwise disjoint positive elements in it has a supremum. The most important examples of extended K-spaces are the following:

( 1) the space M (O , 2 , lt) of equivalence classes of measurable functions, where (O, I, 1r) is a measure space with lt a o-finite measure (or, more generally, a space with the direct sum property, see [18]);

(2) the space C*(Q) of continuous functions defined on an extremally disconnected compact space O with values in the extended real line, taking the values *oo only on a nowhere dense set [4], [9], [55];

(3) the space V of selfadjoint (not necessarily bounded) operators associated with a von Neumann algebra (see [66]).

To save space, we shall restrict attention to the real case, since the analysis of complex K-spaces is entirely analogous. The symbol P(E) will denote the Boolean algebra of order projections in a K-space E . If E contains an order unit, C(E) is the Boolean algebra of unit elements (fragments of the identity) in E. The algebras f@) and €(,8) are isomorphic and known as the base of E. Throughout the sequel, B will be a fixed complete Boolean algebra. The basis for Boolean-valued analyris of vector lattices is the following result. [6]). Let ,9 be the field of real numbers in the model V@) . The algebraic system .q I Q.e., .q with lowered operations and order) is an extended K-space. Moreover, there exists an isomorphism X of the Boolean algebra B onto the base ry@) such that

Theorem (Gordon

forall x,ye .q I and be B.

Throughout the sequel, R will denote the field of real numbers inside V@) . If the base of a K-space E is isomorphic to .8, then E itself is isomorphic to the foundation Eo C.q l, and in this situation ^E is extended if and only if E0 -SE J . Under these circumstances one says that ,% J is a maximal extension and I a Boolean-valued realization of the K-space .E . It is noteworthy that Boolean-valued realizations of certain structures lead to subsystems of the field I .

Theorem [25]. (l)

A subgroup of the additive group of ,q is a Boolean-valued realization of an archimedean lattice-ordered group.

(2) A vector sublattice of ,q , considered as a vector lattice over the field Rî s a Boolean-valued realization of an archimedean vector lattice.

(3) An archimedean f-ring contains two mutually complementary components, one of which is a group with zero multiplication realized as in (l), and the other has a subring of the ring ,9 as a Boolean-valued realization.

(Q An archimedean f -algebra contains two mutually complementary components, one of which is a vector lattice with zero multiplication realized as in (2), and the other is realized as a subring and sublattice of ,q , considered as an f-algebra over R^ .

3.4. Gordon's theorem implies the main structural properties of K-spaces. We shall dwell on the realization of K-spaces and functional calculus. Let Q be a Stonean compact subspace of the Boolean algebra B and define C*(Q) as in 3.1 (2). We call a map e:R --B a resolution of unity in B if (l) e(s) < e(t) (s i r) ; (2) V,.^ e(t) -1, A,.^ e(t) -0 ; (3) V,.,e(s)e(t) (l e R). Let B(R) bethesetof allresolutionsof unityin ^8. Thesets C."(Q) and ^B(R) can be endowed canonically with the structure of an extended Kspace (see [4] and [19]). (a e 91.

3.6. Let gR and ,% (R) be the o-algebra of Borel sets and the vector lattice of Borel functions, respectively, on the real line. We identify B with the algebra of fragments of the identity rn ,9 J (see 3.2). For every x € .q I there exists a unique spectra measure (sequentially o-continuous Boolean homomorphism F:9n -B) such that tt(-x , t) : el (l e n; . The measure p defines an integral I *(f ) :-[ f Al drt(t) (f e ,q 8)).

--JR

In this situation lr(f ) is the unique element of !t I for which [.l',(f) <rnn:p(f</)).

3.7. Theorem (1291, [50]). The map I*:,9(R) + .9/ ! is the unique sequentially o-continuous lattice and algebraic homomorphism satisfying the condition /"(id^)x. then o-limlxoxl : 0 for some x e X. We shall assume that {lxl:x e X}od -E c.q I. lf E is extended, i.e., E :,9 L then X is also said to be extended. An example of an extended Banach-Kantorovich space is the space M(O,Z,lt,Y) of (equivalence classes of) strongly measurable vector-valued functions with values in a Banach space Y.

Theorem 1231. Let x be a

Banach space in the model V@) . Then the lowering x ! is an extended Banach-Kantorovich space. Conversely, if X is an extended Banach-Kantorovich space, there exists a unique (up to linear isometry) Banach space x in V@) whose lowering is linearly isometric to X .

Let us call the bounded part of the space x I the restricted descent of

x. The restricted descents of Banach spaces in V@) constitute the class of .B-cyclic Banach spaces. Let B be the complete Boolean algebra of norm one projections in a Banach space X. We shall say that X is cyclic with respect to B, or .B-cyclic, if, for an arbitrary partition of unity (nr)rr= c B and any bounded family ("r)r.c X there exists a unique element x e X such that nexc: ftex (( e E) and llxll < trp(.= 11x6ll . Let A(B) denote an arbitrary commutative AW* -algebra whose complete Boolean algebra of idempotents is isomorphic to B . lf X is an AW* -model over A(B) (see [52]), then X is a B-cyclic Banach space. All the aforesaid leads to the following realization theorem. [59]. The restricted descent of a complex Hilbert space in the model V@) is an AW* -module over the algebra A(B) . Conversely, for any AW* -module X over A(B) there exists a unique (up to unitary equivalence) Hitbert space inside V@) whose restricted descent is unitarily equivalent to X. 5.5. Let X and Y be Banach-Kantorovich spaces with norming lattices E and f' , respectively. A linear operator T: X -+ Y is said to be majorizable if there exists a positive operator ,S:E --+ F such that lTxl < S(lxl) for all x e X .If E -F and S is an orthomorphism, a majorizable operator is also called bounded, since in that situation 7" coincides with the lowering from V@) of a bounded linear operator acting in Banach spaces. By interpreting Riesz-Schauder theory in a Boolean-valued model one arrives at a new concept of cyclic compactness and obtains corresponding results on the solvability of operator equations in Banach-Kantorovich spaces t241. General majorizable operators have a far more complicated structure and their analysis requires appeal to a considerable variety of methods (see 1241,1261, t3 1 l).

Theorem

5.6. Banach-Kantorovich spaces and majorizable operators were first introduced by L.V. Kantorovich in [6]. It was he who proposed the first applications to the solution of operator equations by the method of successive approximations (see U7l, tl9l). These objects possess a rich structure and have several important applications in the area of spaces of measurable vector-valued functions and linear operators in such spaces t261. In particular, the study of Banach-Kantorovich spaces leads to the notion of Banach spaces with mixed norms, which is enormously useful in connection with the isometric classification of Banach function spaces (see [26]). $6. Banach algebras 6.1. Certain classes of Banach algebras yield some beautiful variations on the theme outlined in the previous section. Call a C*-algebra A a B-C*algebra if A is cyclic with respect to a Boolean algebra of projections B , where any projection in ^B is multiplicative, involutive and of unit norm. If A ts an AW* -algebra and .B a regular subalgebra of the Boolean algebra of central projections p(,4) , then A is a B-C.-algebra. We shatl therefore say that A is a ,B-AW' -algebra if .B is a regular subalgebra of ry c@). Now let A be a J B-algebra and ^B and p.(A) the same as before. lf A is a cyclic Banach space with respect to .B , we shall say that A is a B-J B-algebra. An isomorphism that commutes with the projections in B will be called a B-isomorphism. The following theorem, though in a slightly different form, was proved in 1671. from the model V@) is a B-AW* -algebra (B-J B-atgebra). Conversely, for any B-AW* -algebra (B-J B-algebra) A there exists a unique (to within an isomorphism) AW* -algebra (J B-algebra) ,M whose restricted descent is Bisomorphic to A . In addition, "{ witl be a factor in V@) if and only if B : 9r(A) . The formulated statement concerning AW* -algebras is obtained in l59l and [601. [66], t67l). In particular, it was shown in [59] that for all infinite cardinals a and P there exists an AW* -algebra that is simultaneously aand Phomogeneous (a conjecture of Kaplansky in [52]). This fact is related to the location of cardinal numbers under embeddings in V@) (see [44], t68l). $7. Convex analysis 7.1. The subdifferential is one of the most important concepts in convex analysis (see [24], [28]). In this section, referring to a few examples, we shall show how to use Boolean-valued analysis to study the internal structure of subdifferentials. Take avector space X, d K-space E, and a sublinear operator P: X + E. The subdifferential A P of P at zero is also called the supporting set of P t281. By Gordon's theorem, we may assume that E c 9 L so that we can "convert" P inside a suitable model into an ,9valued sublinear operator, i.e., a sublinear functional. To be precise: 7.2. Theorem [54]. There exist a Banach space x and a continuous sublinear functional p: x -+ ,9 in the model V@) such that there is an isomorphic embedding of X into the Banach-Kantorovich space x I with [(rX) f l,s densein x\-1 and P:Fot.Inthissituation,forany U e?P thereis a unique element u e V@) for which [u e lpn: I and (J : tt J o l. The map U -u is an affine isomorphism of the convex sets 0 P and (0 p) I . 7.3. Thus, the study of 0P largely reduces to that of 0p. For example, let us look at the extremal structure of the subdifferential 0P. Let Ch(P) denote the set of extreme points of 0 P. It should be noted that by Theorem 2 the relations U e Ch(P) and [u e Ch(p)] -1 are equivalent, and one can then use the classical Krein-Mil'man Theorem and Mil'man's inversion of it for 0p. For a rigorous formulation of the result, we need some more definitions. The weak closure o-cl(Q) (cyclic hull mix(A)) is the set of all operators T e L(X , E) of the form Tx : o-hmTox (x e X) , where (7") is a net in C) (resp., Txo-lnrTrx (x e X) , where (f6) c O and (26) is a partition of unity in p(E)) . The weak cyclic closure of Cl is the set o-cl(mix(O)) . If o-clQ -Q or mix(Q) : Q, one says that O is weakly o-closed or cyclic, respectively. The definition of weak r-closedness is analogous. 7.4. Theorem 1271, [281. Q) For any sublinear operator P: X -+ E the subdffirential coincides with the weakly cyclic closure of the convex hull of its extreme points Ch(P) .

(2) If P:X -> E is a sublinear operator and T € L(Y,X), then Ch(PoT)cCh(P) oI.

A set A c L@, ^E) is operator convex (weakly bounded) if oQ+/Q c Q forany o,f eg I-, o*f :1 (theset {Tr:T €O} isorderboundedfor all xeX). 7.5. Theorem 1241, t271. For a weakly bounded set Q c L(X , E) , the following assertions are equivalent:

(1) Cl -0P for some sublinear P: X -. E;

(2) Q rs convex, cyclic, and weakly r-closed;

(3) Q rs convex, cyclic, and weakly o-closed; (4) Q rs operator convex and weakly o-closed.

7.6. Let Q: Z + E be a positive operator, P a sublinear operator from a vector space X to a K-space Z . The term disintegration in K-spaces refers to those parts of the calculus of subdifferentials based on the formula According to Euclid's definition, "a monad is that through which the many become one." In the formal theory, a monad p(q) is defined as an external list of the standard elementsof astandardfilter V,i.e., x e p(7; *-' (V't,F e V)x e F. A syntactic characterization of external sets that are monads was proposed not long ago by Benninghofen and Richter [45]. It is useful to emphasize that everv monad is a union of ultramonads-monads of ultrafilters. For such a monad U the assertions (3x e U)rp(x) and (Vx)p@), where (pe@) is an external formula, are equivalent. Hence it is clear that ultramonads are the genuine "elementary" objects of infinitesimal analysis. 8.2. For applications to the theory of operators, it is of essential importance to construct a synthetic theory in the framework of which both the nonstandard methods offered by Boolean-valued models and external set theories can be used. Only preliminary results have so far been achieved in this direction; they pertain to the study of topological-type notions related with mixing-cyclic filters, topologies and so on, which play major roles in K-spaces. We shall point out one of the possible approaches to cyclic monadology.

8.3. Fix a standard complete Boolean algebra B and an external set A consisting of elements of a separated Boolean-valued universe V@) . An element x e V@) is a member of the cyclic hull mix(,4) if and only if, for some internal family (or)rr= of elements of ,,4 and an internal partition of unity (b)cr= in B, we have

xmix( eeb€x€.A monad p(V) is said to be cyclic if p(V)mix p(,V). A point is said to be essential if it lies in the monad of some pro-ultrafilter-a maximal cyclic filter or, more rigorously, 8n ultrafilter in V@) .