Injective envelopes of C∗-algebras as operator modules

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Nov 2003Embargo end date: 01 Jan 2001 English Publisher:Mathematical Sciences PublishersJournal:Pacific Journal of Mathematics, volume 212, pages 57-69 (issn: 0030-8730,

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Authors: Frank, Michael; Paulsen, Vern I.;

doi: 10.2140/pjm.2003.212.57 , 10.48550/arxiv.math/0106131

arXiv: math/0106131

Injective envelopes of C∗-algebras as operator modules

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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

Keywords

46L08, 46L01, Projective and injective objects in functional analysis, Mathematics - Operator Algebras, multiplier, injective envelope, \(C^*\)-algebra, completely bounded map, injective module, Functional Analysis (math.FA), Mathematics - Functional Analysis, General theory of \(C^*\)-algebras, rigidity, \(C^*\)-modules, FOS: Mathematics, Operator spaces and completely bounded maps, Operator Algebras (math.OA)

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Injective envelopes and local multiplier algebras of C*-algebras
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