When amenable groups have real rank zero $C^{\ast }$-algebras

Abstract

We investigate when discrete, amenable groups have $C^{\ast }$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse in an open problem. We show that if $C^{\ast }(G)$ has real rank zero, then all normal subgroups of $G$ that are elementary amenable and have finite Hirsch length must be locally finite.

arXiv comment13 pages, corrections made from previous version (including ones based on referee's comments). To appear on the International Journal of Mathematics