Isomorphisms of Hilbert C*-Modules and ∗-Isomorphisms of Related Operator C*-Algebras

Let M be a Banach C*-module over a C*-algebra A carrying two A-valued inner products 〈., .〉1, 〈., .〉2 which induce equivalent to the given one norms on M. Then the appropriate unital C*-algebras of adjointable bounded A-linear operators on the Hilbert A-modules {M, 〈., .〉1} and {M, 〈., .〉2} are shown to be ∗-isomorphic if and only if there exists a bounded A-linear isomorphism S of these two Hilbert A-modules satisfying the identity 〈., .〉2 ≡ 〈S(.), S(.)〉1. This result extends other equivalent descriptions due to L. G. Brown, H. Lin and E. C. Lance. An example of two non-isomorphic Hilbert C*-modules with ∗-isomorphic C*-algebras of ”compact”/adjointable bounded module operators is indicated. Investigations in operator and C*-theory make often use of C*-modules as a tool for proving, especially of Banach and Hilbert C*-modules. Impressing examples of such applications are G. G. Kasparov’s approach to Kand KK-theory of C*-algebras [6, 15] or the investigations of M. Baillet, Y. Denizeau and J.-F. Havet [1] and of Y. Watatani [14] on (normal) conditional expectations of finite index on W*-algebras and C*-algebras. In addition, the theory of Hilbert C*-modules is interesting in its own. Our standard sources of reference to Hilbert C*-module theory are the papers [12, 8, 4, 5], chapters in [6, 15] and the book of E. C. Lance [10]. We make the convention that all C*modules of the present paper are left modules by definition. A pre-Hilbert A-module over a C*-algebra A is an A-module M equipped with an A-valued mapping 〈., .〉 : M×M → A which is A-linear in the first argument and has the properties: 〈x, y〉 = 〈y, x〉 , 〈x, x〉 ≥ 0 with equality iff x = 0 . The mapping 〈., .〉 is called the A-valued inner product on M. A pre-Hilbert A-module {M, 〈., .〉} is Hilbert if and only if it is complete with respect to the norm ‖.‖ = ‖〈., .〉‖ A . We always assume that the linear structures of A and M are compatible. One of the key problems of Hilbert C*-module theory is the question of isomorphism of Hilbert C*-modules. First of all, they can be isomorphic as Banach A-modules. But there is another natural definition: Two Hilbert A-modules {M1, 〈., .〉1}, {M2, 〈., .〉2} over a fixed C*-algebra A are isomorphic as Hilbert C*-modules if and only if there exists a bijective bounded A-linear mapping S : M1 → M2 such that the identity 〈., .〉1 ≡ ≡ 〈S(.), S(.)〉2 is valid on M1 × M1. In 1985 L. G. Brown presented two examples of Hilbert C*-modules which are isomorphic as Banach C*-modules but which are nonisomorphic as Hilbert C*-modules, cf. [2, 11, 5]. This result was very surprising since Hilbert space theory, the classical investigations on Hilbert C*-modules like [12, 8], G. G. Kasparov’s approach to KK-theory of C*-algebras relying on countably generated Hilbert C*-modules and other well-known investigations in this field did not give any indication of such a serious obstacle in the general theory of Hilbert C*-modules. L. G. Brown obtained his examples from the theory of different kinds of multipliers of C*-algebras without identity. Furthermore, making use of the results of the Ph.D. thesis of Nien-Tsu Shen [13] he proved the following: For a Banach C*-module M over a C*algebra A carrying two A-valued inner products 〈., .〉1, 〈., .〉2 which induce equivalent to the given one norms on M the appropriate C*-algebras of ”compact” bounded A-linear operators on the Hilbert A-modules {M, 〈., .〉1} and {M, 〈., .〉2} are ∗-isomorphic if and only if there exists a bounded A-linear isomorphism S of these two Hilbert A-modules satisfying 〈., .〉2 ≡ 〈S(.), S(.)〉1, cf. [2, Thm. 4.2, Prop. 4.4] together with [5, Prop. 2.3], ([3]). By definition, the set of ”compact” operators KA(M) on a Hilbert A-module {M, 〈., .〉} is defined as the norm-closure of the set KA(M) of all finite linear combinations of the operators {θx,y : θx,y(z) = 〈z, x〉y for every x, y, z ∈ M}. It is a C*-subalgebra and a two-sided ideal of End∗A(M), the set of all adjointable bounded A-linear operators on {M, 〈., .〉}, what is the multiplier C*-algebra of KA(M) by [8, Thm. 1]. Note, that in difference to the well-known situation for Hilbert spaces, the properties of an operator to be ”compact” or to possess an adjoint depend heavily on the choice of the A-valued inner product on M. These properties are not invariant even up to isomorphic Hilbert structures on M, in general, cf. [5]. We make the convention that operators T which are ”compact”/adjointable with respect to some A-valued inner product 〈., .〉i will be marked T (i) to note where this property arises from. The same will be done for sets of such operators. In 1994 E. C. Lance showed that two Hilbert C*-modules are isomorphic as Hilbert C*-modules if and only if they are isometrically isomorphic as Banach C*-modules ([9]) opening the geometrical background of this functional-analytical problem and extending a central result for C*-algebras: C*-algebras are isometrically multiplicatively isomorphic if and only if they are ∗-isomorphic, [7, Thm. 7, Lemma 8]. At the contrary, non-isomorphic Hilbert structures on a given Hilbert A-module M over a C*-algebra A can not appear at all if M is self-dual, i. e. every bounded module map r : M → A is of the form 〈., ar〉 for some element ar ∈ M (cf. [4, Prop. 2.2,Cor. 2.3]), or if A is unital and M is countably generated, i. e. there exists a countably set of generators inside M such that the set of all finite A-linear combinations of generators is norm-dense in M (cf. [2, Cor. 4.8, Thm. 4.9] together with [6, Cor. 1.1.25] and [5, Prop. 2.3]). Now, we come to the goal of the present paper: Whether for a Banach C*-module M over a C*-algebra A carrying two A-valued inner products 〈., .〉1, 〈., .〉2 which induce equivalent to the given one norms on M the appropriate C*-algebras End A (M) and End (2,∗) A (M) of all adjointable bounded A-linear operators on M are ∗-isomorphic, or not? This question is non-trivial since even non-∗-isomorphic non-unital C*-algebras can possess a common multiplier C*-algebra: For example, on the closed interval [0, 2] ⊂ R consider the C*-algebra of all continuous functions vanishing at zero together with the C*-algebra of all continuous function vanishing at one. They are non-∗-isomorphic, but the multiplier C*-algebra C([0,2]) of them consisting of all continuous functions on [0,2] is the same in both cases. That is, additional arguments are needed to describe the relation between the multiplier C*-algebras of non-∗-isomorphic C*-algebras of ”compact” operators on some Banach C*-modules carrying non-isomorphic C*-valued inner products. One quickly realizes that the techniques of multiplier theory are not suitable to shed some more light on this general situation. One has to turn back to C*-theory and to the properties of ∗-isomorphisms, as well as to the theory of Hilbert C*-modules. Theorem: Let A be a C*-algebra and M be a Banach A-module carrying two A-valued inner products 〈., .〉1, 〈., .〉2 which induce equivalent to the given one norms. Then the following conditions are equivalent: (i) The Hilbert A-modules {M, 〈., .〉1} and {M, 〈., .〉2} are isomorphic as Hilbert C*modules. (ii) The Hilbert A-modules {M, 〈., .〉1} and {M, 〈., .〉2} are isometrically isomorphic as Banach A-modules. (iii) The C*-algebras K (1) A (M) and K (2) A (M) of all ”compact” bounded A-linear operators on both these Hilbert C*-modules, respectively, are ∗-isomorphic. (iv) The unital C*-algebras End (1,∗) A (M) and End (2,∗) A (M) of all adjointable bounded Alinear operators on both these Hilbert C*-modules, respectively, are ∗-isomorphic. Further equivalent conditions in terms of positive invertible quasi-multipliers of K (1) A (M) can be found in [5]. Proof. The equivalence of (i) and (ii) was shown by E. C. Lance [9], and the equivalence of (i) and (iii) turns out from a result for C*-algebras of L. G. Brown [2, Thm. 4.2, Prop. 4.4] in combination with [5, Prop. 2.3]. Referring to G. G. Kasparov [8, Thm. 1] the implication (iii)→(iv) yields naturally. Now, suppose the unital C*-algebras End (1,∗) A (M) and End (2,∗) A (M) are ∗-isomorphic. Denote this ∗-isomorphism by ω. One quickly checks that the formula x ∈ M → 〈x, x〉1,Op. = θ (1) x,x ∈ K (1) A (M) defines a K (1) A (M)-valued inner product on the Hilbert A-module M regarding it as a right K (1) A (M)-module. Moreover, the set {K(x) : x ∈ M, K ∈ K (1) A (M)} is norm-dense inside M since the limit equality x = ‖.‖M − lim n→∞ (θ x,x(θ (1) x,x + n ))(x) holds for every x ∈ M. As a first step we consider the intersection of the two C*-subalgebras and two-sided ideals ω(K (1) A (M)) and K (2) A (M) inside the unital C*-algebra End (2,∗) A (M). The intersection of them is a C*-subalgebra and two-sided ideal of End (2,∗) A (M) again. It contains the operators θ x,y · ω(θ (1) z,t ) = θ (2) ω(θ (1) z,t )(x),y = θ (2)