Invariant trace simplices and relative property (T)
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914511268610048 |
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| author | Slutsky, Raz |
| author_facet | Slutsky, Raz |
| contents | Let $α\colon G\curvearrowright A$ 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 $C_r^*(Δ\wr G)$ has Bauer trace simplex for every countable discrete group $Δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24738 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Invariant trace simplices and relative property (T) Slutsky, Raz Operator Algebras Functional Analysis Group Theory 46L55 Let $α\colon G\curvearrowright A$ 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 $C_r^*(Δ\wr G)$ has Bauer trace simplex for every countable discrete group $Δ$. |
| title | Invariant trace simplices and relative property (T) |
| topic | Operator Algebras Functional Analysis Group Theory 46L55 |
| url | https://arxiv.org/abs/2604.24738 |