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

TL;DR

This paper analyzes Hamana’s injective envelope $I(A)$ for a unital $C^*$-algebra $A$ in the operator-space setting, introducing completely bounded ($cb$) $A$-module maps and establishing a rigidity principle that extends beyond positive maps. It proves that $I(A)$ is the minimal tight $A$-injective extension (in left, right, and bimodule senses) and that cb $A$-module maps extend to multiplications by elements of $I(A)$, enabling a natural embedding of multipliers and of the local multiplier algebras $M_{loc}(A)$. It then develops the theory of relative injectivity and tightness, connecting these notions to projections and cohomology, and shows how local and quasi-multipliers sit inside $I(A)$ and can be identified with injective envelopes of the corresponding local structures. The results unify the treatment of injective envelopes, multipliers, and regular completions in the operator-space context and yield new, simpler proofs for known projection and multiplier results.

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.

Theorems & Definitions (32)

• Theorem 2.1
• proof
• Corollary 2.2: Rigidity
• proof
• Definition 1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Definition 2