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.
