Journal of the Australian Mathematical Society - Series A Vol. 64 Part 3 (1998) 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-1ex(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 $A\times_\alpha G$ 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. PDF file size: 96K © Copyright 1998, Australian Mathematical Society