On reduced twisted group C*-algebras that are simple and/or have a unique trace

TL;DR

This paper investigates conditions under which reduced twisted group C*-algebras are simple or have a unique trace, introducing the relative Kleppner condition as a key tool and applying it to various groups.

Contribution

It introduces the relative Kleppner condition as a new criterion for simplicity and unique trace properties in twisted group C*-algebras, expanding understanding of these algebras for different groups.

Findings

Established new sufficient conditions for simplicity and unique trace.

Applied conditions to wreath products and Baumslag-Solitar groups.

Demonstrated the effectiveness of the relative Kleppner condition.

Abstract

We study the problem of determining when the reduced twisted group C*-algebra associated with a discrete group G is simple and/or has a unique tracial state, and present new sufficient conditions for this to hold. One of our main tools is a combinatorial property, that we call the relative Kleppner condition, which ensures that a quotient group G/H acts by freely acting automorphisms on the twisted group von Neumann algebra associated to a normal subgroup H. We apply our results to different types of groups, e.g. wreath products and Baumslag-Solitar groups.