On Function and Operator Modules

Let A be a unital Banach algebra. We give a characterization of the left Banach A-modules X for which there exists a commutative unital C-algebra C(K), a linear isometry i:X → C(K), and a contractive unital homomorphism θ:A → C(K) such that i(a· x) = θ(a)i(x) for any a ∈ A, x ∈ X . We then deduce a “commutative” version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a w-version of the latter characterization, which generalizes some previous work of Effros and Ruan. 1991 Mathematics Subject Classification. 46H25, 46J10, 47D25, 46B28. 1 Introduction. Let H be a Hilbert space and let B(H) be the C-algebra of all bounded operators on H. Let A ⊂ B(H) and B ⊂ B(H) be two unital closed subalgebras and let X ⊂ B(H) be a closed subspace. If axb belongs to X whenever a ∈ A, x ∈ X , b ∈ B, then X is called a (concrete) operator A-B-bimodule. The starting point of this paper is the abstract characterization of these bimodules due to Christensen, Effros, and Sinclair. Namely, let us consider two unital operator algebras A and B and let X be an arbitrary operator space. Assume that X is an A-B-bimodule. Then for any integer n ≥ 1, the Banach space Mn(X) of n×n matrices with entries in X can be naturally regarded as an Mn(A)-Mn(B)bimodule, by letting [aik]· [xkl]· [blj] = [ ∑ 1≤k,l≤n aik· xkl· blj ] for any [aik] ∈ Mn(A), [xkl] ∈ Mn(X), [blj ] ∈ Mn(B). It is shown in [CES] that if ‖a· x· b‖ ≤ ‖a‖‖x‖‖b‖ for any n ≥ 1 and any a ∈ Mn(A), x ∈ Mn(X), and b ∈ Mn(B), then there exist a Hilbert space H, and three complete isometries (1.1) J :X → B(H), π1:A → B(H), π2:B → B(H), such that π1, π2 are homomorphisms and J(a· x· b) = π1(a)J(x)π2(b) for any a ∈ A, x ∈ X , b ∈ B. In that case, X is called an (abstract) operator A-B-bimodule. Equivalently (in the