Multipliers of Operator Spaces, and the Injective Envelope

We study the injective envelope I(X) of an operator space X, showing amongst other things that it is a self-dual C∗module. We describe the diagonal corners of the injective envelope of the canonical operator system associated with X. We prove that if X is an operator A-B-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the operator algebras that can act on X are determined by these corners of I(X) and consequently bimodule actions on X extend naturally to actions on I(X). These results give another characterization of the multiplier algebra of an operator space, which was introduced by the first author, and a short proof of a recent characterization of operator modules, and a related result. As another application, we extend Wittstock’s module map extension theorem, by showing that an operator A-B-bimodule is injective as an operator A-B-bimodule if and only if it is injective as an operator space.