Reduced twisted crossed products over C*-simple groups

TL;DR

This paper establishes a correspondence between ideals and tracial states in reduced twisted crossed products over C*-simple groups, characterizing simplicity and uniqueness of traces via properties of the underlying algebra.

Contribution

It introduces a bijective correspondence between ideals and tracial states in reduced twisted crossed products over C*-simple groups, extending understanding of their structure.

Findings

Bijective correspondence between maximal ideals and invariant ideals.

Equivalence of simplicity of the crossed product and the underlying algebra's invariant ideals.

Characterization of unique tracial states via invariant tracial states.

Abstract

We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and the reduced crossed product has a unique tracial state if and only if the underlying C*-algebra has a unique invariant tracial state. We also show that the reduced crossed product satisfies an averaging property analogous to Powers' averaging property.