Journal of the Australian Mathematical Society (Series A)

Research Article

Extremally rich C*-crossed products and the cancellation property

Ja A Jeonga1 and Hiroyuki Osakaa2

a1 Department of Mathematics Kyung Hee University Seoul, 130 - 701, Korea

a2 Mathematics Institute University of Copenhagen Universitetsparken 5 DK-2100, Copenhagen Ø Denmark and Department of Mathematics Sciences Ryukyu University Nishihara-cho, Okinawn 903 - 01, Japan

Abstract

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 xα G 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.

(Received May 12 1997)

(Revised October 21 1997)

1991 Mathematics subject classification (Amer. Math. Soc.)

  • 46L05