Group $C^*$-algebras of locally compact groups acting on trees

TL;DR

This paper investigates specific $C^*$-algebras associated with groups acting on trees, revealing their structure, relationships, and uniqueness properties, especially in the context of $L^p$-integrability and boundary actions.

Contribution

It characterizes the structure of $L^p$-group $C^*$-algebras for groups acting on trees, proving their uniqueness under certain conditions and analyzing quotient maps between them.

Findings

Quotient maps between $L^{p+}$ and $L^{q+}$ $C^*$-algebras are not injective for certain groups.

$L^{p+}$ $C^*$-algebras are the only $C^*$-algebras from $G$-invariant ideals in $B(G)$ under specific conditions.

Any $C^*$-algebra distinguishable from the universal one and with dual space as a $G$-invariant ideal is isomorphic to the reduced $C^*$-algebra.

Abstract

We study the group $C^*$-algebras $C^*_{L^{p+}}(G)$ - constructed from $L^p$-integrability properties of matrix coefficients of unitary representations - of locally compact groups $G$ acting on (semi-)homogeneous trees of sufficiently large degree. These group $C^*$-algebras lie between the universal and the reduced group $C^*$-algebra. By directly investigating these $L^p$-integrability properties, we first show that for every non-compact, closed subgroup $G$ of the automorphism group $\mathrm{Aut}(T)$ of a (semi-)homogeneous tree $T$ that acts transitively on the boundary $\partial T$ and every $2 \leq q < p \leq \infty$, the canonical quotient map $C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ is not injective. This reproves a result of Samei and Wiersma. We prove that under the additional assumptions that $G$ acts transitively on $T$ and that it has Tits' independence property, the group $C^*$-algebras $C^*_{L^{p+}}(G)$ are the only group $C^*$-algebras coming from $G$-invariant ideals in the Fourier-Stieltjes algebra $B(G)$. Additionally, we show that given a group $G$ as before, every group $C^*$-algebra $C^*_{\mu}(G)$ that is distinguishable (as a group $C^*$-algebra) from the universal group $C^*$-algebra of $G$ and whose dual space $C^*_\mu(G)^*$ is a $G$-invariant ideal in $B(G)$ is abstractly ${}^*$-isomorphic to the reduced group $C^*$-algebra of $G$.