Injective Envelopes of $C^*$-algebras as Operator Modules

Abstract

In this paper we give some characterizations of M. Hamana's injective envelope I(A) of a C*-algebra A in the setting of operator spaces and completely bounded maps. These characterizations lead to simplifications and generalizations of some known results concerning completely bounded projections onto C*-algebras. We prove that I(A) is rigid for completely bounded A-module maps. This rigidity yields a natural representation of many kinds of multipliers as multiplications by elements of I(A). In particular, we prove that the(n times iterated) local multiplier algebra of A embeds into I(A). Some remarks on local left/right/quasi multiplier algebras as subsets of I(A) are added.