Injective envelopes of C^∗-algebras as operator modules

Michael Frank and Vern I. Paulsen

Vol. 212 (2003), No. 1, 57–69

DOI: 10.2140/pjm.2003.212.57

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).

Milestones

Received: 26 April 2000

Revised: 26 June 2001

Published: 1 November 2003