Murray-von Neumann equivalence of projections in C*-algebras
by
Lawrence G. Brown
Purdue University

There are several cancellation properties for Murray-von Neumann equivalence of projections in C^*-algebras. The stable version of strong cancellation, p \oplus r ~ q \oplus r implies p ~ q, can be equivalently stated: [p] = [q] in K₀(A) implies p ~ q, where p and q can be in M_n(A). The strong cancellation property follows from stable rank one and is equivalent to stable rank one when A has real rank zero. Weak cancellation is that if p and q generate the same ideal I and have the same image in K₀(I), then p ~ q. This property was studied jointly with G. Pedersen and is related to extremal richness (a generalization of stable rank one). An intermediate cancellation property is that if p and q generate the same ideal and have the same image in K₀(A), then p ~ q. This cancellation property is related to the property (IR) introduced by Friis and Rordam. A natural problem is to find conditions under which these properties will carry over from A to M(A). A modest partial result: Assume A has stable rank one and is the direct limit of an increasing sequence of ideals. If each M(I_n) has the cancellation property, then so does M(A).

Date received: June 18, 2002