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

TL;DR

This paper characterizes M. Hamana's injective envelope of a C*-algebra within operator spaces, simplifying known results and revealing new insights into multipliers and local multiplier algebras as operator modules.

Contribution

It provides new characterizations of the injective envelope as an operator module, leading to generalizations and a better understanding of multipliers in C*-algebras.

Findings

Injective envelope is rigid for completely bounded A-module maps.

Local multiplier algebra embeds into the injective envelope.

Simplified characterizations of multipliers as elements of the injective envelope.

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.