Journal of Operator Theory

Volume 44, Issue 2, Fall 2000 pp. 413-431.

Extremal richness of multiplier and Corona algebras of simple $C^*$-algebras with real rank zero

Authors: Francesc Perera
Author institution: Departament de Matematiques, Universitat Autonoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

Summary: In this paper we investigate the extremal richness of the multiplier algebra $\mul$ and the corona algebra $\mul/A$, for a simple \C\ $A$ with real rank zero and stable rank one. We show that the space of extremal quasitraces and the scale of $A$ contain enough information to determine whether $\mul/A$ is extremally rich. In detail, if the scale is finite, then $\mul/A$ is extremally rich. In important cases, and if the scale is not finite, extremal richness is characterized by a restrictive condition: the existence of only one infinite extremal quasitrace which is isolated in a convex sense.

Keywords: Extremal richness, real rank, stable rank, refinement monoid

Contents Full-Text PDF