Detecting ideals in reduced crossed product C*-algebras of topological dynamical systems

TL;DR

This paper introduces the $ ext{l}^1$-ideal intersection property for crossed product C*-algebras, linking it to C*-simplicity and C*-uniqueness, and demonstrates its presence in various complex group actions.

Contribution

It establishes the $ ext{l}^1$-ideal intersection property for broad classes of topological dynamical systems, extending previous results on C*-uniqueness to twisted groupoid algebras.

Findings

The $ ext{l}^1$-ideal intersection property holds for systems with lattices in Lie groups.

It applies to linear groups over number fields and virtually polycyclic groups.

Extended results on C*-uniqueness of $ ext{L}^1$-groupoid algebras to twisted settings.

Abstract

We introduce the $\ell^1$-ideal intersection property for crossed product C*-algebras. It is implied by C*-simplicity as well as C*-uniqueness. We show that topological dynamical systems of arbitrary lattices in connected Lie groups, arbitrary linear groups over the integers in a number field and arbitrary virtually polycyclic groups have the $\ell^1$-ideal intersection property. On the way, we extend previous results on C*-uniqueness of $\mathrm{L}^1$-groupoid algebras to the general twisted setting.