Invariant trace simplices and relative property (T)

TL;DR

This paper explores conditions under which the simplex of invariant traces for a group action on a C*-algebra is Bauer, extending the Glasner-Weiss theorem to noncommutative settings.

Contribution

It establishes a noncommutative analogue of the Glasner-Weiss theorem using relative property (T) and ergodicity conditions, with applications to various group actions.

Findings

If (G,H) has relative property (T) and ergodicity holds, then the invariant trace simplex is Bauer.

Criteria are provided for ergodicity in specific group action contexts.

For groups with property (T) and trivial amenable radical, certain reduced crossed products have Bauer trace simplices.

Abstract

Let $\alpha\colon G\curvearrowright A$ be an action of a countable discrete group on a separable unital $C^*$-algebra. We study the simplex $\mathrm{T}(A)^G$ of $G$-invariant traces and ask when it is Bauer. Our main result is a noncommutative version of the Glasner-Weiss theorem: if $(G,H)$ has relative property (T) and the $H$-action on the von Neumann algebra of every extremal invariant trace is ergodic, that is, has only scalar fixed points, then $\mathrm{T}(A)^G$ is Bauer. We give criteria for the ergodicity hypothesis and apply them to certain quasi-local permutation actions, generalized Bernoulli actions, traces on group $C^*$-algebras, and reduced crossed products. In particular, if $G$ is infinite, has property (T), and trivial amenable radical, then $C_r^*(\Delta\wr G)$ has Bauer trace simplex for every countable discrete group $\Delta$.