Group $C^{\ast }$-algebras of locally compact groups acting on trees

Abstract

We study the group $C^{\ast }$-algebras $C_{L^{p+}}^{\ast }(G)$ - 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 $C^{\ast }$-algebras lie between the universal and the reduced group $C^{\ast }$-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 $2\le q<p\le \infty$, the canonical quotient map $C_{L^{p+}}^{\ast }(G)\twoheadrightarrow C_{L^{q+}}^{\ast }(G)$ 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 $C^{\ast }$-algebras $C_{L^{p+}}^{\ast }(G)$ are the only group $C^{\ast }$-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 $C^{\ast }$-algebra $C_{\mu }^{\ast }(G)$ that is distinguishable (as a group $C^{\ast }$-algebra) from the universal group $C^{\ast }$-algebra of $G$ and whose dual space $C_{\mu }^{\ast }(G{)}^{\ast }$ is a $G$-invariant ideal in $B(G)$ is abstractly ${}^{\ast }$-isomorphic to the reduced group $C^{\ast }$-algebra of $G$.

v2: 27 pages, minor modifications

References in corpus (1)

• On the ideal structure of the Fourier-Stieltjes algebra of certain groups