Some algebraic methods in real rank zero

Ken Goodearl, University of California Santa Barbara

We will introduce and discuss two algebraic concepts - the Crawley-Jónsson `exchange property' and the `separative cancellation' condition - in the context of C*-algebras with real rank zero. The first of these properties enters the picture via the following theorem: A C*-algebra A has real rank zero if and only if the regular module has the exchange property. One consequence is that the stable rank of A (assuming real rank zero) is determined by cancellation conditions within the monoid V(A) (the Murray-von Neumann equivalence classes of projections from $M_\infty(A)$). Separativity is the condition ; it seems to be satisfied for every `known' (or at least every well-understood) C*-algebra with real rank zero. Among its consequences is that the stable rank is restricted to the values 1, 2, $\infty$. We conclude by establishing the separative case of a conjecture of Shuang Zhang: If A is a separative C*-algebra with real rank zero, then $K_1(A)\cong U(A)/U(A)^\circ$. (That $U(A)/U(A)^\circ$ embeds in K₁(A) is due to Huaxin Lin.) The results mentioned are from joint work with P. Ara, K.C. O'Meara, E. Pardo and R. Raphael.

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