Extremally rich C^*-crossed products
and the cancellation property

Ja A Jeong
Department of Mathematics
Kyung Hee University
Seoul, 130 - 701
Korea

and
Hiroyuki Osaka
Mathematics Institute
University of Copenhagen
Universitetsparken 5
DK-2100, Copenhagen Ø
Denmark
and
Department of Mathematical Sciences
Ryukyu University
Nishihara-cho, Okinawa 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^-1_q) is dense in A, where $\ex(A)$ is the set of extreme points in the closed unit ball A₁ of A. In [#!bp2!#,#!bp3!#] Brown and Pedersen introduced this notion and showed that A is extremally rich if and only if $\conv(\ex(A))=A_{1}$. 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 $A\times_\alpha G$ 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 $\sr(A)=1$, and has the SP-property, then it is shown that any crossed product $A\times_\alpha G$ 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.

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