EXTREMALLY RICH C-ASTERISK-CROSSED PRODUCTS AND THE CANCELLATION PROPERTY

A unital C-algebra A is called extremally rich if the set of quasi-in vertible elements A(-1) ex(A)A(-1) (= A(q)(-1)) is dense in A, where e x(A) is the set of extreme points in the closed unit ball A(1) of A. I n [7, 8] Brown and Pedersen introduced this notion and showed that A i s extremally rich if and only if conv(ex(A)) = Al. Any unital simple C -algebra with extremal richness is either purely infinite or has stab le rank one (sr(A) = 1). In this note we investigate the extremal rich ness of C-crossed products of extremally rich C*-algebras by finite g roups. It is shown that if A is purely infinite simple and unital then A x(alpha) G is extremally rich for any finite group G. But this is n ot true in general when G is an infinite discrete group. If A is simpl e with sr(A) = 1, and has the SP-property, then it is shown that any c rossed product A x(alpha) G by a finite abelian group G has cancellati on. Moreover if this crossed product has real rank zero, it has stable rank one and hence is extremally rich.