arXiv:1409.1343v1 [math.OA] 4 Sep 2014 Kreı̆n C*-modules Paolo Bertozzinia , Kasemsun Rutamornb a Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Bangkok 12121, Thailand e-mail: paolo.th@gmail.com b Department of Mathematics, Faculty of Education, Dhonburi Rajabhat University, Bangkok 10600, Thailand e-mail: kasemamorn270@hotmail.com 10 December 2013∗ Abstract We introduce a notion of Kreı̆n C*-module over a C*-algebra and more generally over a Kreı̆n C*-algebra. Some properties of Kreı̆n C*-modules and their categories are investigated. Keywords: Kreı̆n space, Kreı̆n C*-module, tensor product. MSC-2010: 47B50, 46C20, 46L08, 46L87, 46M15, 53C50. Contents 1 Introduction 1 2 Kreı̆n C*-Algebras 2 3 Kreı̆n C*-modules over C*-algebras 3 4 Kreı̆n C*-modules over Kreı̆n C*-algebras 7 5 Categories of Kreı̆n C*-modules 10 6 Outlook 12 1 Introduction Vector spaces with an indefinite inner product started to appear in physics with the work on special relativistic space-time by H.Minkowski [M] and were later used for the first time in quantum field theory by P.Dirac [D] and W.Pauli [P], but their first mathematical discussion was provided by L.Pontrjagin [Po] and since then they have been an object of study mainly of the Russian school. Kreı̆n spaces, i.e. complete vector spaces equipped with an indefinite inner product, were formally defined by Ju.Ginzburg [Gi] and in their present form by E.Scheibe [Sc]. Their properties have been investigated by several mathematicians such as I.Iohvidov, H.Langer, R.Phillips, M.Naı̆mark, M.Kreı̆n ∗ This is a reformatted version for arXiv of a paper published in Chamchuri Journal of Mathematics (2013) 5:23-44. 1 and his school and have been extensively used in quantum field theory via the Gupta-Bleuler [B, G] formalism in quantum electrodynamics.1 They have been reconsidered in quantum field theory by K.Kawamura [K1, K2] who also proposed axioms for Kreı̆n C*-algebras (involutive algebras of bounded linear operators on a Kreı̆n space). Kreı̆n spaces also appeared prominently in the definition of semi-Riemannian spectral triples in non-commutative geometry, by A.Strohmaier [Str], M.Paschke, A.Sitarz [PS] and more recently by K.van den Dungen, M.Paschke, A.Rennie [DPR]. Hilbert C*-modules (complete modules over a C*-algebra with a C*-algebra-valued positive inner product) are a generalization of Hilbert spaces where the field of complex numbers is substituted by a general C*-algebra. They were first introduced in 1953 by I.Kaplanski [K] in the case of commutative unital C*-algebras. Between 1972 and 1974 W.Paschke [Pa1, Pa2] and M.Rieffel [R1, R2, R3] extended the theory to the case of modules over arbitrary C*-algebras and after that the subject grew and spread rapidly. The purpose of this paper is to introduce an extremely elementary notion of Kreı̆n C*-module over a Kreı̆n C*-algebra, generalizing to the module case the usual decomposability condition of a Kreı̆n space into its “positive” and “negative” subspaces, and closely following the definition of Kreı̆n C*-module over a C*-algebra elaborated in S.Kaewumpai [Ka]. In practice, such “decomposable” kind of Kreı̆n modules admit a non-canonical splitting as direct sums of Hilbert C*-modules, with opposite signature that, when we allow the algebra to be a Kreı̆n C*-algebra, can be chosen to be “compatible” with one of the fundamental symmetry automorphisms of the algebra. We first recall in section 2 (a variant of) the definition of Kreı̆n C*-algebra by K.Kawamura [K1, K2] and, in section 3, for the benefit of the reader, we reproduce in some detail the key definitions and proofs of the main results on Kreı̆n C*-modules over C*-algebras that were developed in S.Kaewumpai’s thesis [Ka]. In section 4, we further extend the previous definition to cover the case of Kreı̆n C*-modules over Kreı̆n C*-algebras. In the subsequent section 5 we expand the notion of tensor product of Kreı̆n C*-modules over C*-algebras, formulated in R.Tanadkithirun [T] in order to cover our more general situation and we discuss some of the properties of the categories of modules and bimodules so obtained. Several examples illustrating the scope of the definitions are presented. Of particular interest are the possible applications to the spectral geometry of semi-Riemannian manifolds and their noncommutative counterparts. As it can be clearly appreciated by these geometric examples of modules of vector fields over the Clifford algebra of a semi-Riemannian manifold, the notion of Kreı̆n C*-module that is contained here is very specific and corresponds to the special case of tangent bundles admitting a global decomposition as Whitney orthogonal sums of positive and negative definite Hermitian vector sub-bundles i.e. semiRiemannian manifolds that are time-orientable and space-orientable.2 More general notions of Kreı̆n C*-modules are necessary to deal with cases where such global “splitting” is not available, but for now we do not enter this interesting discussion that will very likely also require modifications in the definition of Kreı̆n C*-algebra. 2 Kreı̆n C*-Algebras The following is a variation of the definition of Kreı̆n C*-algebra introduced by K.Kawamura [K1]. Definition 2.1. A Kreı̆n C*-algebra is an involutive complete complex topological algebra (i.e. a complete complex topological vector space with a bilinear continuous product and a continuous involution) A that admits at least one fundamental symmetry i.e. an involutive automorphism 1 See as main references: J.Bognar[Bo], T.Azizov, I.Iohvidov [AI], M.Dritschel, J.Rovnyak [DR], E.Kissin, V.Shulman [KS]. 2 For details on semi-Riemannian geometry we refer to B.O’Neill [O]. 2 α : A → A with α ◦ α = ιA and one Banach algebra norm k · kα (inducing the given topology) such that kα(a∗ )akα = kak2α , for all a ∈ A. Proposition 2.2 (K. Kawamura, Example 2.4, Section 2.3). The set B(K) of linear continuous operators on a Kreı̆n space K is a Kreı̆n C*-algebra. Every fundamental symmetry J of a Kreı̆n space K is associated to a fundamental symmetry a 7→ JaJ , a ∈ B(K) of the Kreı̆n C*-algebra B(K). Remark 2.3. Note that although, contrary to K.Kawamura, we assume the existence of a given topology, we do not fix a priori any Banach norm on the Kreı̆n C*-algebra, so that several topologically equivalent Banach norms can exist. Specifically, for every fundamental symmetry α there is a unique norm k · kα making A a C*-algebra, denoted by Aα , that coincides with A as a complex algebra and whose involution is given by x†α := α(x∗ ), for all x ∈ A. For example, different fundamental symmetries of a Kreı̆n space K, induce operator norms on the Kreı̆n C*-algebra B(K) of bounded linear operators on K that do not coincide, although they are topologically equivalent. y In the subsequent sections, we will often use the notation A+ for the even part of the Kreı̆n C*-algebra A under a fundamental symmetry α, i.e. the C*-algebra of elements such that α(x) = x; and similarly the notation A− for the odd part of the Kreı̆n C*-algebra A under a fundamental symmetry α i.e. the Hilbert C*-module, over A+ , of elements such that α(x) = −x. Later on we will see natural situations motivated from semi-Riemannian geometry that seem to require further generalization of the definition of Kreı̆n C*-algebra, but for this work we will mostly limit our consideration to the definition above. 3 Kreı̆n C*-modules over C*-algebras In this section, we recall some basic material on unital Kreı̆n C*-modules over unital C*-algebras that was developed in S.Kaewumpai’s Master thesis [Ka]. The material naturally covers, as a special case, the situation of Kreı̆n spaces (that are Kreı̆n C*-modules over the C*-algebra C) and will be further generalized in the subsequent section where we will consider modules over Kreı̆n C*-algebras. Recall that, given a unital right module EA over a unital C*-algebra A, an A-valued Hermitian inner product is a map h· | ·i : E × E → A such that, for all x, y, z ∈ E, a ∈ A: hz | x + yi = hz | xi + hz | yi, hz | xai = hz | xia, hx | yi∗ = hy | xi. In the case of unital left modules A E, the second property above is substituted with hax | zi = ahx | zi. Whenever confusion might arise, we will denote an inner product on EA by h· | ·iE and an inner product on A E by E h· | ·i. The direct sum E ⊕ F of two right (left) unital modules, over the unital C*-algebra A, equipped with the inner product defined by hx1 ⊕ y1 | x2 ⊕ y2 iE⊕F := hx1 | x1 iE + hy1 | y2 iF , for all x1 , x2 ∈ E and y1 , y2 ∈ F, is a right (left) unital module over A called orthogonal direct sum of EA and FA . A Hermitian inner product is positive if hx | xi ∈ A> := {a∗ a | a ∈ A}, for all x ∈ E, where A> denotes the positive part of the unital C*-algebra A. An inner product is non-degenerate if hx | xi = 0 ⇒ x = 0A . A unital right (left) Hilbert C*-module E over the unital C*-algebra A is a unital right (left) module on the unital C*-algebra A, that is equipped with an A-valued positivepnon-degenerated inner product, making it a complete metric space with respect to the norm kxk := khx | xikA . The family L(EA ) of A-linear operators on the right (left) unital A-module EA is a complex algebra with multiplication given by composition of maps. If the modules EA and FA are equipped with A-valued inner products, we say that a map T : E → F is adjointable if there exists another map 3 S : F → E such that hy | T (x)iF = hS(y) | xiE , for all x ∈ E and y ∈ F. If the inner product is non-degenerate, adjointable maps are necessarily unique and A-linear and, denoting by T ∗ the unique adjoint of T we have, for Hermitian inner products, (T ∗ )∗ = T , (T ◦ S)∗ = S ∗ ◦ T ∗ and (T1 + αT2 )∗ = T1∗ + αT2∗ so that the family B(EA ) of adjointable operators is an involutive complex subalgebra of L(EA ). In the case of Hilbert C*-modules, adjointable maps are always continuous and the algebra B(EA ) is always a unital C*-algebra when equipped with the operator norm.3 Definition 3.1. If EA is module with an inner product h· | ·i over the C*-algebra A, its antimodule, here denoted by −EA , is the A-module EA equipped with the opposite inner product. −h· | ·i. Definition 3.2. A unital right Kreı̆n C*-module KA over the unital C*-algebra A is a unital right module over A with an A-valued inner product such that KA is isomorphic to the “indefinite” orthogonal direct sum KA = K+A ⊕ K−A of a Hilbert C*-module K+A , with the antimodule K−A of a Hilbert C*-module −K−A . Unital left Kreı̆n C*-modules are similarly defined. Any such decomposition of a Kreı̆n C*-module over A will be called a fundamental decomposition. Fundamental decompositions are in general not unique. Definition 3.3. To every fundamental decomposition KA = K+A ⊕ K−A of a right unital Kreı̆n C*-module KA over the unital C*-algebra A there is an associated fundamental symmetry operator J : KA → KA given by J(x) := x+ − x− , where x = x+ + x− is the direct sum decomposition of x ∈ KA ; and there is an associated Hilbert C*-module |K|A := K+A ⊕ (−K−A ). Making use of the unicity of the expression of vectors as sums of components belonging to the direct summands of a fundamental decomposition and of the definition of the fundamental symmetry associated to such decomposition, we obtain the following statement. Proposition 3.4. Let KA = K+A ⊕ K−A be a fundamental decomposition of a Kreı̆n A-C*-module KA , its fundamental symmetry operator J : KA → KA satisfies the following properties for all x, y ∈ K and a ∈ A: J(x + y) = J(x) + J(y), J(xa) = J(x)a, J ◦ J = IdK , hJ(x) | yiK = hx | J(y)iK , or equivalently hJ(x) | J(y)iK = hx | yiK , ±h(IdK ±J)(x) | (IdK ±J)(x)iK ≥ 0. Every map J : K → K that satisfies the previous list of properties is the fundamental symmetry of a unique fundamental decomposition, denoted by K = KJ+ ⊕ KJ− , where KJ± := {x ∈ K | J(x) = ±x}. The operators (IdK ±J)/2 are a pair of orthogonal projections onto KJ± . The inner product in the Kreı̆n C*-module KA = KJ+A ⊕ KJ−A and the inner product in the associated Hilbert C*-module |K|JA := KJ+A ⊕ (−KJ−A ) are related by hx | yiK = hJ(x) | yi|K|J , hJ(x) | yiK = hx | yi|K|J , ∀x, y ∈ K. Theorem 3.5. If KA is a unital right (left) Kreı̆n C*-module over the unital C*-algebra A and J1 , J2 are two fundamental symmetries of K with associated fundamental decompositions K = KJ+1 ⊕ KJ−1 3 For all the details of these standard results on Hilbert C*-modules, the reader can consult N.Landsman [L, L1], E.Lance [La], N.Wegge-Olsen [WO] or S.Kaewunpai [Ka]. 4 and K = KJ+2 ⊕ KJ−2 , there are two A-linear continuous bijective maps of Hilbert A-C*-modules T+J2 J1 : KJ+1 → KJ+2 and T−J2 J1 : −(KJ−1 ) → −(KJ−1 ), given by T+J2 J1 (xJ+1 ) := xJ+2 and T−J2 J1 (xJ−1 ) := xJ−2 , where x = xJ+1 + xJ−1 and x = xJ+2 + xJ−2 are the direct sum decompositions of x ∈ KA in each of the fundamental decompositions. Proof. The map T+J2 J1 : KJ+1 → KJ+2 is adjointable with adjoint given by the map T+J1 J2 : KJ+2 → KJ+1 , defined by J1 T+J1 J2 (y) := y+ , for y ∈ KJ+2 , J2 since hT+J2 J1 (x) | yiKJ2 = hxJ+2 | y+ iKJ2 = hx | yiK = hxJ+1 | yiKJ1 = (x | T+J1 J2 (y))KJ1 , for all + + + + x ∈ KJ+1 and all y ∈ KJ+2 . In exactly the same way we have that T−J2 J1 : KJ−1 → KJ−2 is adjointable J1 with adjoint given by the map T−J1 J2 : KJ−2 → KJ−1 defined by T−J1 J2 (y) := y− , for all y ∈ KJ−2 . Since an adjointable operator between Hilbert C*-modules (and also between anti-Hilbert C*-modules) is necessarily continuous, both T+J2 J1 , T−J2 J1 and their adjoints T+J1 J2 , T−J1 J2 are continuous A-linear maps. J1 J2 If x, y ∈ KJ+1 and T+J2 J1 (x) = T+J2 J1 (y), we have xJ+1 − y+ = x − y = xJ−2 − y− and since we have J2 J2 J2 J2 0 ≤ hx − y | x − yiKJ1 = hx − y | x − yiK = hx− − y− | x− − y− iKJ2 ≤ 0, via the non-degeneracy of + the Kreı̆n C*-module KA , we obtain the injectivity of T+J2 J1 . − J1 Let (xn ) be a sequence in K+ such that T+J2 J1 (xn ) converges to a given point z ∈ Im(T+J2 J1 ) ⊂ KJ+2 . Since h(xn − xm )J−2 | (xn − xm )J−2 iK ≤ 0, we have kxn − xm k2KJ1 = hxn − xm | xn − xm iK + ≤ hT+J2 J1 (xn − xm ) | T+J2 J1 (xn − xm )iK = kT+J2 J1 (xn ) − T+J2 J1 (xm )k2KJ2 , + J1 so that (xn ), being a Cauchy sequence in the Hilbert C*-module KJ+1 , converges to a point xo ∈ K+ . J2 J1 J2 J1 J2 J1 J1 J2 By continuity of T+ we get z = limn→∞ T+ (xn ) = T+ (xo ) ∈ Im(T+ ), that provides the closure of the range of T+J2 J1 . The fact that T+J2 J1 : KJ+1 → KJ+2 is an adjointable map between Hilbert C*-modules with closed range is equivalent (see for example N.Wegge-Olsen [WO, corollary 15.3.9]) to the complementability of the submodule Im(T+J2 J1 ) ⊂ KJ+2 . If y ∈ Im(T+J2 J1 )⊥ ⊂ KJ+2 , for all x ∈ KJ+1 , we get hT+J1 J2 (y) | xiKJ1 = hy | T+J2 J1 (x)iKJ2 = 0 and + + hence y ∈ Ker(T+J1 J2 ). Since T+J1 J2 is injective, we have Im(T+J2 J1 )⊥ = {0} and since Im(T+J2 J1 ) is complementable, from KJ+2 = Im(T+J2 J1 ) ⊕ Im(T+J2 J1 )⊥ = Im(T+J2 J1 ), we obtain the surjectivity of T+J2 J1 . Theorem 3.6. If KA is a unital right (left) Kreı̆n C*-module, over A, with two fundamental symmetries J1 and J2 with associated direct sum decompositions K = KJ+1 ⊕ KJ−1 = KJ+2 ⊕ KJ−2 , then the Hilbert C*-modules |K|J1 := KJ+1 ⊕ KJ−1 and |K|J2 := KJ+2 ⊕ KJ−2 have equivalent norms. Hence on KA there is a natural topology called the strong topology. Proof. By theorem 3.5, we have two bijective A-linear adjointable (hence continuous) maps T+J2 J1 : KJ+1 → KJ+2 , T−J2 J1 : −KJ−1 → −KJ−2 . It follows immediately that their direct sum map T+J2 J1 ⊕ T−J2 J1 : |K|J1 → |K|J2 is a bijective linear continuous map with continuous inverse and hence as Banach spaces |K|J1 and |K|J2 have equivalent norms. 5 Proposition 3.7. Let KA = KJ+ ⊕ KJ− be the fundamental decomposition of the right (left) unital Kreı̆n C*-module KA associated to the fundamental symmetry J and with associated Hilbert C*-module |K|JA . The operator T : KA → KA is an adjointable operator in KA if and only if it is adjointable in |K|JA . The adjoint T ∗ of T in the Kreı̆n C*-module KA and the adjoint T †J of T in the Hilbert C*-module |K|JA are related by the following formulas: T †J = J ◦ T ∗ ◦ J, T ∗ = J ◦ T †J ◦ J. The family B(KA ) of adjointable operators in the unital right (left) Kreı̆n C*-module KA coincides as a set with the C*-algebra B(|K|JA ) of adjointable operators in the unital right (left) Hilbert C*-module |K|A . Proof. If T is adjointable in the Kreı̆n C*-module KA with adjoint T ∗ , we have, for all x, y ∈ KA , hx | T (y)iK = hT ∗ (x) | yiK and so hJ(x) | T (y)i|K|J = hJ(T ∗ (x)) | yi|K|J and taking x := J(z) we get hz | T (y)i|K|J = hJ ◦ T ∗ ◦ J(z) | yi|K|J that gives the adjointability of T in |K|JA with adjoint T †J = J ◦ T ∗ ◦ J. Following the same passages in the reverse order establishes the equivalence of the notions of adjointability in KA and in |K|JA . Theorem 3.8. The algebra B(KA ) of adjointable endomorphisms of a unital right (left) Kreı̆n C*-module KA is a Kreı̆n-C*-algebra. Proof. By the previous proposition, we see that B(KA ) = B(|K|JA ) as sets and also as unital associative algebras, since the operations of addition and scalar multiplication are the same. Taking on B(KA ) the operator norm k · kJ defined in the C*-algebra B(|K|JA ) of the unital Hilbert C*-module |K|JA , we see that B(KA ) is a Banach space. The involution on the algebra B(KA ) is obtained via the Kreı̆n C*-module adjoint T 7→ T ∗ . In order to complete the proof that B(KA ) is a Kreı̆n C*-algebra, we need to provide an involutive automorphism α : B(KA ) → B(KA ) such that kα(T ∗ ) ◦ T kJ = kT k2J for all T ∈ B(KA ). For this purpose, for all T ∈ B(KA ), we define α(T ) := J ◦ T ◦ J and, using the C*-property of B(|K|JA ), verify that kα(T ∗ ) ◦ T kJ = k(J ◦ T ◦ J) ◦ T kJ = kT †J ◦ T kJ = kT k2J . Proposition 3.9. Let KA be a unital right (left) Kreı̆n C*-module. Any two fundamental symmetries J1 , J2 ∈ B(KA ) of K are unitarily equivalent. Proof. With the notation used in theorem 3.5, consider the adjointable operator U := T+J2 J1 ⊕ T−J2 J1 and note that U ◦ J1 = J2 ◦ U with U ∗ = U −1 . Example 3.10. Every Kreı̆n-space is a Kreı̆n-C*-module over the C*-algebra C, the complexification MC := M⊗R C of Minkowski space in special relativity being probably the most important example. y Example 3.11. Let M be a semi-Riemannian manifold that for now (although this makes the situation less interesting for applications to physics) we suppose to be compact. If the manifold is also supposed to be space-orientable and time-orientable (for a specific example, consider T2 with the indefinite metric coming from the product of a copy of T with positive metric and another copy with negative metric), the module Γ(T (M )) of continuous sections of its tangent bundle T (M ) is a unital Kreı̆n C*-module over the unital C*-algebra C(M ) of continuous functions. Note that, when a semi-Riemannian manifold is not space-orientable and time-orientable, although each fiber of the tangent bundle T (M ) is still a Kreı̆n space, the module Γ(T (M )) of continuous vector fields, fails to be a Kreı̆n C*-module over the C*-algebra C(M ), because in this case Γ(T (M )) does not admit a global splitting as a direct sum of a Hilbert and an anti-Hilbert C*-module (equivalently the tangent bundle T (M ) is not a Whitney direct sum of a positive definite and a negative definite sub-bundle). As a consequence, this very interesting kind of complete semi-definite Hilbert C*-modules do not admit a globally defined fundamental symmetry! y 6 4 Kreı̆n C*-modules over Kreı̆n C*-algebras Definition 4.1. A left Kreı̆n C*-module over a Kreı̆n C*-algebra A is a complete topological vector space that is also a left (unital) module K over A with the the following properties: • K is equipped with an A-valued inner product A (x + y | z) = A (x | z) + A (y | z), A (· | ·) : K × K → A such that ∀x, y, z ∈ K, A (ax | y) = a · A (x | y), ∀x, y ∈ K, ∀a ∈ A, ∗ ∀x, y ∈ K, A (x | y) = A (y | x), ∀y ∈ K, A (x | y) = 0 ⇒ x = 0, • there exists a fundamental symmetry JA : K → K such that J ◦ J = IdK , JA (x + y) = JA (x) + JA (y), ∀x, y ∈ K, JA (a · x) = α(a) · JA (x), ∀x ∈ K, ∀a ∈ A, α(A (x | y)) = A (JA (x) | JA (y)), ∀x, y ∈ K, • for the given choice of fundamental symmetries α on A and JA on K, we have that the map (x, y) 7→ A (x | JA (y)) gives to K the structure of a Hilbert C*-module over the C*-algebra Aα whose norm induces the original topology of K. In a perfectly similar way, there is a definition of right Kreı̆n C*-module K over a Kreı̆n C*-algebra B, where the B-valued inner product (· | ·)B : K × K → B satisfies (x | yb)B = (x | y)B · b, for all b ∈ B and all x, y ∈ K; the fundamental symmetry JB satisfies JB (x · b) = JB (x) · β(b), for all x ∈ K and b ∈ B; and where all the other properties remain essentially unchanged. The following definition of bimodule, whenever we further impose the additional fullness requirements A = A (K | K) := span{A (x | y) | x, y ∈ K} and B = (K | K)B := span{(x | y)B | x, y ∈ K}, extends to the Kreı̆n C*-algebras context the usual notion of imprimitivity Hilbert C*-bimodule that provides the well-known (strong) Morita equivalence of C*-algebras (see also remark 5.7). Definition 4.2. A Kreı̆n C*-bimodule A KB is a left Kreı̆n C*-module over A that is the same time a right Kreı̆n C*-module over B with additional properties: (a · x) · b = a · (x · b), JA = JB and with right and left inner products related via: A (x | y)x = x(y | x)B , for all x, y ∈ K. The compatibility condition requested above on the inner products assures that the induced left and right norms on the Hilbert C*-module K coincide. Remark 4.3. In our definitions the two auxiliary Aα -valued Hilbert C*-module inner products (x | y)JAAα := (x | JA (y))A , JA (x | y)Aα := (JA (x) | y)A , can be used in place of each other since they are related by the isomorphism α of the C*-algebra Aα : α((JA (x) | y)A ) = (x | JA (y))A . y Remark 4.4. Note that KB becomes naturally a Kreı̆n C*-module over the C*-algebra B+ when equipped with the even part of the original inner product i.e. taking 21 (x | y)B + 21 (J(x) | J(y))B as inner product on K; however with this new inner product the submodules K+ and K− are orthogonal (as in our original definition of Kreı̆n C*-module over a C*-algebra) contrary to the general situation here, where the obstruction to the orthogonality is measured by the odd part of the original inner product 21 (J(x) | y)B + 21 (x | J(y))B that is always a non-degenerate anti-Hermitian product on K 7 with values in B− . Furthermore, although in some situations (for example in finite dimensional cases) the product topology induced by the decomposition K = K+ ⊕ K− (that is actually the topology induced by the even part of the inner product, since the two inner products coincide, modulo sign, on their restriction to K+ and K− respectively) is the same as the original topology of K, we still suspect that in general this might fail. A significant difference from the case of Kreı̆n C*-modules over C*-algebras is that the fundamental symmetry JB is in general not self-adjoint (and in general not even adjointable, as can be seen in the case of example 4.7 below, where adjointability happens only in the case of α equal to the identity i.e. when A is already a C*-algebra) with respect to the original Kreı̆n inner product as suggested by the general lack of orthogonality between the even and odd submodules K+ and K− . y Example 4.5. Every Kreı̆n C*-module K over a C*-algebra A, as defined in the previous section, is a special case of our new definition as results by taking α to be the identity isomorphism of A. Actually, whenever the Kreı̆n C*-algebra A is a C*-algebra and α is trivial, we reduce to the definition of the last section: the submodules K+ and K− are orthogonal and the fundamental symmetry JA is Hermitian. The new definition of Kreı̆n module over a Kreı̆n C*-algebra even allows for a possible choice of a nontrivial α also in the case of a C*-algebra A and in this situation we obtain a Kreı̆n C*-module over a C*-algebra where the two submodules K+ and K− fail to be orthogonal and JA is not Hermitian. y Example 4.6. Let KA be a right Kreı̆n C*-module over the C*-algebra A and let B(KA ) be the Krein C*-algebra of adjointable operators on K, then B(K) K is a left Kreı̆n C*-module over the Kreı̆n C*-algebra B(KA ) with left inner product defined by B(K) (x | y) := Θx,y , where Θx,y (z) := x·(y | z)A , for all x, y, z ∈ K. The bimodule B(K) KA is actually a Kreı̆n C*-bimodule such that B(K) (x | y)z = x(y | z)A . for all x, y, z ∈ K. y Example 4.7. Let A be a Kreı̆n C*-algebra. Then A A and AA are both Kreı̆n C*-modules with inner products given by (x | y)A := x∗ y and A (x | y) := xy ∗ , furthermore A AA is a Kreı̆n C*-bimodule. y Example 4.8. Let K1 and K2 be two Kreı̆n spaces. The space B(K2 ) B(K1 , K2 )B(K1 ) of linear continuous maps between them is a Kreı̆n C*-bimodule with the left/right actions given by the usual compositions of linear operators and inner products given respectively by (T | S)B(K1 ) := T ∗ ◦ S and B(K2 ) (T | S) := T ◦ S ∗ . y Example 4.9. Following the definitions provided in [BR], let A, B ∈ ObA be two objects in a Kreı̆n C*-category A . Then AAB := HomA (B, A) is a Kreı̆n C*-bimodule over the Kreı̆n C*-algebras AAA on the left and ABB on the right. y Example 4.10. Let M be Minkowski space (or more generally any real vector space equipped with semi-definite inner product); let ΛC (M) denote the space of complex-valued antisymmetric forms on M (the complexified Grassmann algebra of M) and let Cl(M) denote the complexified Clifford algebra of M. Note that for every fundamental decomposition of M = M+ ⊕ M− , we have for the Grassmann ˆ C (M− ) and similarly, for the Clifford algebras, algebras the decomposition ΛC (M) := ΛC (M+ )⊗Λ ˆ Cl(M− ), where (if we work in the category of associative algebras) ⊗ ˆ denotes the Cl(M ) = Cl(M+ )⊗ Z2 -graded tensor product. Note that the (underlying complex vector space of the) Grassmann algebra ΛC (M) is naturally a Kreı̆n space with the semi-definite inner product induced by universal factorization property via M C C (ω1 ∧ · · · ∧ ωn | φ1 ∧ · · · φn ) := det[(ωi | φj )] on each of the summands in ⊕dim q=0 Λq (M) = Λ (M), C where (ωi | φj ) ∈ C denotes the Minkowski inner product on complexified covectors ωi , φj ∈ Λ1 (M). 8 The Clifford algebra Cl(M) has a natural structure of Keı̆n C*-algebra as a sub-algebra of the Kreı̆n C*-algebra B(ΛC (M)): every fundamental symmetry JM of Minkowski space lifts to an involutive automorphims of Cl(M) (by the defining universal property of the Clifford algebra) that, under the linear isomorphism Cl(M) ≃ ΛC (M), coincides with the (second quantized) fundamental symmetry ∧q C C JΛC (M) := ⊕∞ q=0 JM of the Kreı̆n space Λ (M). It follows that the Kreı̆n space Λ (M) is a left Kreı̆n module over the Kreı̆n C*-algebra Cl(M). The (underlying vector space of the) Grassmann algebra ΛC (M) also becomes a Kreı̆n C*-bimodule over Cl(M) via Clifford left and right actions and with the inner products induced via the linear isomorphism ΛC (M) ≃ Cl(M) by the standard Kreı̆n C*-bimodule structure of the Kreı̆n C*-algebra Cl(M) over itself. As described in more detail in H.Baum [Ba] (see also A.Strohmaier [Str, section 5.1] and K.Van Den Dungen-M.Paschke-A.Rennie [DPR, section 3.3.1]) the module S(M) of (Dirac) spinors is a Kreı̆n space, whose fundamental symmetries are proportional to the product of the operators of Clifford multiplication by all the vectors in an orthonormal basis for the timelike summand of a fundamental decomposition M = M+ ⊕ M− .4 The space S(M) (for M even-dimensional) becomes a left Kreı̆n C*-module over the Keı̆n C*-algebra Cl(M) with the inner product induced by the linear isomorphisms S(M) ⊗ S(M)∗ ≃ ΛC (M) ≃ Cl(M) and Cl(M) S(M)C is a Kreı̆n C*-bimodule that, with the terminology introduced in remark 5.7, is an example of Morita-Kreı̆n equivalence C*-bimodule. y Example 4.11. Let M be a (compact) semi-Riemannian space-orientable and time-orientable manifold. As already described in example 3.11, the module Γ(T (M )) of its continuous vector fields is a (unital) Kreı̆n C*-bimodule over the (unital) C*-algebra C(M ). The algebra Γ(Cl(M )) of continuous section of the complexified Clifford bundle Cl(T (M )) of M is a (unital) Kreı̆n C*-algebra and the module Γ(ΛC (M )) of continuous sections of the complexified Grassmann bundle ΛC (M ) of M is a (unital) Kreı̆n C*-bimodule over the Kreı̆n C*-algebra Γ(Cl(M )). The case of spinorial manifolds is described in example 5.9. y Remark 4.12. Note that, in the previous example, if the manifold M is not time-orientable and spaceorientable, the algebra Γ(Cl(M )) (although being a nice involutive complete topological algebra) does not admit a globally defined fundamental symmetry and so does not fit into the current definition of Kreı̆n C*-algebra! This clearly indicates that the environment of Kreı̆n C*-algebras and Kreı̆n C*-modules that we have developed here is insufficient to deal with a general axiomatization of “complete semi-definite C*-algebras and C*-modules over them”. y We pass now to briefly examine the main properties of the algebras of adjointable operators on Kreı̆n C*-modules over Kreı̆n C*-algebras. Definition 4.13. Let KA be a Kreı̆n C*-module over the Kreı̆n C*-algebra A. A map T : K → K is said to be adjointable if there exists another map T ∗ : K → K such that (T (x) | y)A = (x | T ∗ (y))A , for all x, y ∈ K. The family of such adjointable maps is denoted by B(KA ). Remark 4.14. As usual, the adjointable maps are already A-linear and continuous and the adjoint T ∗ is unique. The set B(KA ) is a vector space and an associative unital algebra under composition, furthermore the map ∗ : T 7→ T ∗ is involutive, antimultiplicative and conjugate C-linear so that B(KA ) is a complex associative unital ∗-algebra. y Proposition 4.15. A map T : KA → KA is adjointable with respect to the inner product (· | ·)A if and only if the map T is adjointable for the Hilbert C*-module KAα with the auxiliary inner product 4 On the usual Minkowski space M4 , the Kreı̆n space S(M4 ) has signature (2, 2) and the fundamental symmetries are just the Dirac γ 0 operators. 9 (· | ·)JAAα . As a consequence, the associative unital algebra B(KA ) coincides with the associative unital algebra B(KAα ). The relation between the adjoint T ∗ of T in B(KA ) and the adjoint T †α of T in the C*-algebra B(KAα ) is given by: T †α = JA ◦ T ∗ ◦ JA , T ∗ = JA ◦ T †α ◦ JA Proof. Suppose that (T (x) | y)A = (y | T ∗ (y))A . The following calculation (T (x) | y)JAAα = (T (x) | JA (y))A = (x | T ∗ JA (y))A = (x | JA JA T ∗ JA (y))A = (x | JA T ∗ JA (y))JAAα , assures that the adjointability in B(KA ) implies the adjointability in B(KAα ) and the first formula relating the adjoints. Suppose now that (T (x) | y)JAAα = (x | T †α (y))JAAα i.e. (T (x) | JA (y))A = (x | JA T †α (y))A and choosing y = JA (y ′ ) for an arbitrary y ′ ∈ K, we obtain (T (x) | (y ′ ))A = (x | JA T †α JA (y ′ ))A that assures the reverse and the second adjointability formula. Although we know that in general JA is not an adjointable operator, we still have the following fundamental symmetry of B(KA ): Proposition 4.16. If T is adjointable in B(KA ), also the new operator JA ◦ T ◦ JA is adjointable in B(KA ) and the map αJA : T 7→ JA ◦ T ◦ JA is a ∗-isomorphism of the involutive algebra B(KA ) of adjointable operators. Proof. Since, for all T ∈ B(KA ), (T ∗ )∗ = T , we obtain JA (JA T †α JA )†α JA = T or equivalently (JA T †α JA )†α = JA T JA . Since (S †α )†α = S, we get (JA T †α JA ) = (JA T JA )†α i.e. αJA is a †α -isomorphism of the C*-algebra B(KAα ). Similarly from (S †α )†α = S, we get JA (JA S ∗ JA )∗ JA = S or equivalently (JA S ∗ JA )∗ = JA SJA and hence JA S ∗ JA = (JA SJA )∗ i.e. αJA is a ∗-isomorphism of the involutive algebra B(KA ). Theorem 4.17. The algebra B(KA ) of adjointable operators of a Kreı̆n C*-module over a Kreı̆n C*-algebra A is a Kreı̆n C*-algebra. Proof. The ∗-isomorphism αJA of the involutive algebra B(KA ), defined in the previous proposition, satisfies the C*-property kαJA (T ∗ )T kα = kT k2α with respect to the norm of the C*-algebra B(KAα ). 5 Categories of Kreı̆n C*-modules The following proposition, whose proof is self-evident, provides the most elementary category of morphisms of Kreı̆n C*-algebras that naturally contains, as a full subcategory, the category of unital ∗-homomorphims of unital C*-algebras. Proposition 5.1. There is a category A whose objects are unital Kreı̆n C*-algebras A, B, . . . ; whose arrows φ : A → B are unital Kreı̆n ∗-homomorphisms i.e. unital ∗-homomorphisms of involutive unital algebras φ : A → B such that there exist at least a fundamental symmetry of α of A and β of B such that φ ◦ α = β ◦ φ; and composition is the usual composition of functions. We now define the Kreı̆n C*-analogue of the well-known notion of C*-correspondence. 10 Definition 5.2. A left Kreı̆n C*-correspondence from the unital Kreı̆n C*-algebra B to unital the Kreı̆n C*-algebra A is unital left Kreı̆n C*-module A M over the Kreı̆n C*-algebra A equipped with a morphism of unital Kreı̆n C*-algebras from B to the unital Kreı̆n C*-algebra B(A M) of adjointable operators on the Kreı̆n module A M such that x · β(b) = JA (JA (x) · b). A right Kreı̆n C*-correspondence from B to A is similarly defined as a unital right Kreı̆n C*-module NB over B equipped with a morphism of unital Kreı̆n C*-algebras from A to B(NB ) such that α(a) · x = JB (a · (JB x)). A morphism of Kreı̆n C*-correspondences is a map Φ : A MB → A NB between right (respectively left) Kreı̆n C*-correspondences such that Φ(a · x · b) = a · Φ(x) · b, ∀a ∈ A, b ∈ B, x ∈ M. Remark 5.3. The previous definition entails that a left Kreı̆n C*-correspondence is actually a unital bimodule A MB over the the unital Kreı̆n C*-algebras A and B (with A-valued inner product) such that there exists at least one fundamental symmetry J of A M and fundamental symmetries α of A, β of B that satisfy the compatibility condition J(axb) = α(a)J(x)β(b), for all x ∈ M, a ∈ A and b ∈ B. Clearly we have categories of morphisms of right (respectively) Kreı̆n C*-correspondences under the usual composition of morphisms. y The following definition and theorems incorporate (and generalize to the case of Kreı̆n C*-modules over Kreı̆n C*-algebras) the notion of tensor product of Kreı̆n spaces and Kreı̆n C*-modules over C*-algebras developed in R.Tanadkithirun’s senior undergraduate project [T]. Definition 5.4. The internal tensor product of two right Kreı̆n C*-correspondences A MB and B NC is defined as a left A-linear right C-linear and B-balanced map ⊗ : A MB × B NC → A TC with values into a right Kreı̆n C*-correspondence A TB from A to B such that the following universal factorization property is satisfied: for every left A-linear right C-linear and B-balanced function φ : M × N → Q with values into a Kreı̆n C*-correspondence A QC from A to C, there exists a unique morphism Φ : T → Q of Kreı̆n C*-correspondences such that Φ ◦ ⊗ = φ. Theorem 5.5. Tensor products af right Kreı̆n C*-correspondences exist and are unique up to isomorphism in the category of morphisms of Kreı̆n C*-correspondences. Proof. The unicity up to isomorphism is a standard consequence of a definition via universal factorization properties. For the proof of existence, consider the fundamental decompositions M = M+ ⊕ M− and N = N+ ⊕ N− induced by a pair of fundamental symmetries JM and JN that are compatible with three fundamental symmetries α of A, β of B and γ of C. Using the algebraic tensor product of bimodules and the canonical isomorphism of bimodules (M+ ⊕ M− ) ⊗B (N+ ⊕ N− ) ≃ [(M+ ⊗B N+ ) ⊕ (M− ⊗B N− )] ⊕ [(M+ ⊗B N− ) ⊕ (M− ⊗B N+ )], (5.1) we have that JM ⊗B JN is a fundamental symmetry of M⊗B N that induces the previous decomposition and is compatible with the left action of A and the right action of C. By universal factorization property (two times), we can define on M ⊗B N a unique C-valued inner product such that, for all x1 , x2 ∈ M and y1 , y2 ∈ N, N BN (x1 ⊗B y1 | x2 ⊗B y2 )M⊗ := (y1 | (x1 | x2 )M B · y2 )C . C The following property holds for this inner product γ((x1 ⊗B y1 | x2 ⊗B y2 )C ) = ((JM ⊗B JN )(x1 ⊗B y1 ) | (JM ⊗B JN )(x2 ⊗B y2 ))C 11 and the algebra A acts by adjointable operators on the left. The inner product so defined on M ⊗B N induces on each one of the direct summands of the even part of the decomposition in formula (5.1) the structure of Hilbert C*-module and, for the summands of the odd part, the structure of anti-Hilbert C*-module over the same C*-algebra (C, †γ ). Theorem 5.6. There is a weak category M• whose objects are unital Kreı̆n C*-algebras A, B, . . . ; whose arrows are right Kreı̆n C*-correspondences; and whose composition is obtained by internal tensor product of Kreı̆n C*-correspondences. In a totally similar way, we have a weak category • M of left Kreı̆n C*-correspondences under internal tensor product. Proof. The associativity of composition modulo isomorphism is assured using the universal factorization property. The (weak) identities are given the Kreı̆n C*-algebras A considered as right Kreı̆n C*-correspondences over themselves when equipped with their standard right inner product (a1 | a2 )A := a∗1 a2 , a1 , a2 ∈ A. Remark 5.7. The previous categories M• and • M are actually 2-categories considering as 2-arrows the morphisms of Kreı̆n C*-correspondences with their usual functional composition as composition over 1-arrows and their internal tensor product as composition over objects. This pair of weak 2-categories is the “Kreı̆n counterpart” to the usual 2-categories of right and left C*-correspondences and their (common) subcategory of 1-isomorphisms, that consists of Kreı̆n C*-bimodules A MB that are full and satisfy the imprimitivity condition A (x | y)z = x(y | z)B , for all x, y, z ∈ M, is the “Kreı̆n counterpart” of the Morita-Rieffel weak category of imprimitivity C*-bimodules that describe the (strong) Morita equivalence between C*-algebras5 and hence we have a theoretical background capable of discussing the notion of “Kreı̆n-Morita equivalence” at least in the context of Kawamura’s Kreı̆n C*-algebras. y Remark 5.8. The previous categories (exactly as their C*-counterparts) are not equipped with involutions: the contragredient of a right correspondence is a left correspondence, but usually not another right correspondence.6 More interesting notions of “bivariant” Kreı̆n C*-bimodules will be developed elsewhere. y Example 5.9. Whenever the time-orientable space-orientable (compact) semi-Riemannian evendimensional manifold M admits a spinorial structure, or more generally a spinc structure, (see details in H.Baum [Ba]) the family Γ(S(M )) of continuous section of a given complex spinor bundle S(M ) becomes a Kreı̆n-Morita equivalence Kreı̆n C*-bimodule between C(M ) (on the right) and the Kreı̆n C*-algebra Γ(Cl(M )) on the left.7 Its contragredient Kreı̆n C*-bimodule Γ(S(M ))∗ is isomorphic to the Kreı̆n C*-bimodule of sections of the dual spinor bundle S(M )∗ and we have Γ(S(M )) ⊗C(M) Γ(S(M ))∗ ≃ Γ(ΛC (M )) as tensor product of Kreı̆n C*-bimodules. y 6 Outlook The discussions of duality and of spectral theory, via suitable “Kreı̆n bundles”, for some “commutative” subclasses of the Kreı̆n C*-modules here defined, will be dealt with in future works.8 5 For additional details on the Morita-Rieffel categories and strong Morita equivalence see for example the review sections in [BCL] and the references therein. 6 Recall that, given a bimodule A KB , over involutive algebras A, B, its contragredient bimodule B KA is the same Abelian group K := K with left/right actions defined via b · x · a := a∗ xb∗ , for all a ∈ A, b ∈ B, x ∈ K. For a right Kreı̆n C*-correspondence KB with right inner product (x | y)B , for x, y ∈ K, the contragredient B K is naturally a left Kreı̆n C*-correspondence via B (x | y) := (x | y)B , for all x, y ∈ K, A similar statement holds for left correspondences. 7 A similar statement holds in the odd-dimensional case if the Clifford algebra Γ(Cl(M )) is replaced by its even part Γ(Cl+ (M )), see [GVF, section 9.2]. 8 See anyway [BBL] for some elementary results in the case of commutative Kreı̆n C*-algebras. 12 The notion of Kreı̆n C*-module over a Kreı̆n C*-algebra that we presented here, although interesting as a first step to explore some of the issues in semi-definite situations, is still too elementary to be fully useful for general applications to non-commutative spectral geometry, at least whenever the semiRiemannian geometry involved presents topological obstructions to orientability, either in spacelike or in timelike sense (or both). Since global fundamental symmetries in Kreı̆n C*-algebras are remnants of the fundamental decompositions of the Kreı̆n spaces on which they are faithfully represented, their existence in situations coming from semi-Riemannian geometry seems to be a consequence of such global topological conditions of orientability and it is likely that a more general definition of a complete semi-definite analog of C*-algebras might be necessary to deal with such cases. A possible line of attack would be to define semi-definite modules that are direct summand submodules of our “freesplitting” Kreı̆n modules over a C*-algebra (eliminating the topological obstruction on orientability via “embedding” into a wider environment exactly as we usually do in the case of projective but non-free modules) and redefine Kreı̆n C*-algebras as compressions of the “free-splitting” Kawamura case. We might explore these and other possibilities in subsequent work. A more immediately achievable important goal (especially in view of applications to examples of semiRiemannian geometries related to relativistic physics) is the removal of the unitality (compactness) requirements in the definitions of Kreı̆n C*-modules and Kreı̆n C*-algebras. Our main long-term interest is to formulate notions of semi-definite involutive operator algebraic environments that are suitable, as a (topological) background, for the development of non-commutative geometry and spectral triples in a completely general semi-definite situation. Notes and acknowledgments The paper originates from and further elaborates on material presented in S. Kaewunpai master thesis [Ka] as well as on R. Tanadkithirun and A.Atchariyabodee undergraduate senior projects [T, A]. Thanks to Starbucks Coffee at the 1st floor of the Emporium Suites Tower for the welcoming environment where this research work has been discussed and written. The authors thank the two anonymous referees of the paper. 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