Injective Envelopes of $C^{\ast }$-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.

LaTeX2e (amsart). to appear in Pacific J. Math

References in corpus (1)

• On Hahn-Banach type theorems for Hilbert C*-modules

Cited by in corpus (8)

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• Characterizations of essential ideals as operator modules over C*-algebras
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• When is the second local multiplier algebra of a C*-algebra equal to the first?
• Injective Envelopes and Local Multiplier Algebras of Some Spatial Continuous Trace C*-Algebras
• Injective Envelopes of Separable C*-algebras
• The second local multiplier algebra of a separable C*-algebra