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Extremally rich C*-crossed products and the cancellation property

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

Ja A Jeong  and
Hiroyuki Osaka
Ja A Jeong
Affiliation:
Department of Mathematics Kyung Hee University Seoul, 130 - 701, Korea
Hiroyuki Osaka
Affiliation:
Mathematics Institute University of CopenhagenUniversitetsparken 5 DK-2100, Copenhagen ØDenmark and Department of Mathematics Sciences Ryukyu UniversityNishihara-cho, Okinawn 903 - 01, Japan
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Abstract

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A unital C*-algebra A is called extremally rich if the set of quasi-invertible elements A-1 ex (A)A-1 (= A-1q) is dense in A, where ex(A) is the set of extreme points in the closed unit ball A1 of A. In [7, 8] Brown and Pedersen introduced this notion and showed that A is extremally rich if and only if conv(ex(A)) = A1. Any unital simple C*-algebra with extremal richness is either purely infinite or has stable rank one (sr(A) = 1). In this note we investigate the extremal richness of C*-crossed products of extremally rich C*-algebras by finite groups. It is shown that if A is purely infinite simple and unital then A xα, G is extremally rich for any finite group G. But this is not true in general when G is an infinite discrete group. If A is simple with sr(A) =, and has the SP-property, then it is shown that any crossed product A xαG by a finite abelian group G has cancellation. Moreover if this crossed product has real rank zero, it has stable rank one and hence is extremally rich.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 3 , June 1998 , pp. 285 - 301
Copyright
Copyright © Australian Mathematical Society 1998

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