Multipliers of operator spaces, and the injective envelope

We study the injective envelope I(X) of an operator space X, showing amongs t other things that it is a self-dual module. We describe the diagonal corn ers of the injective envelope of the canonical operator system associated w ith 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 algebr a of an operator space, which was introduced by the rst author, and a short proof of a recent characterization of operator modules, and a related resu lt. As another application, we extend Wittstock's module map extension theo rem, 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.