Invariant trace simplices and relative property (T)

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 $\Delta$.