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One-sided ideals and approximate identities in operator algebras

Part of: Linear spaces and algebras of operators Topological algebras, normed rings and algebras, Banach algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

David P. Blecher
David P. Blecher
Affiliation:
Department of Mathematics, University of Houston, 4800 Calhoun, Houston, TX 77204-3008, USA e-mail: dblecher@math.uh.edu
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Abstract

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A left ideal on any C*-algebra is an example of an operator algebra with a right contractive approximate indentiy (r.c.a.i.). Indeed, left ideal in C*-algebras may be charcterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. shoulod be studied in terms of a certain let ideal of a C*-algebra. We study left ideals from the perspective of ‘Hamana theory’ and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approzimate indentity, including a Banach-Stone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 46L07: Operator spaces and completely bounded maps 47L30: Abstract operator algebras on Hilbert spaces 46H10: Ideals and subalgebras 47L75: Other nonselfadjoint operator algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 425 - 448
Copyright
Copyright © Australian Mathematical Society 2004

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