Group C*-Algebras of Locally Compact Groups Acting on Trees

Abstract

Abstract It was proved by Samei and Wiersma that for every non-compact, closed subgroup $G$ of the automorphism group $\textrm {Aut}(T)$ of a (semi-)homogeneous tree $T$ acting transitively on the boundary $\partial T$ and every $2 \leq q < p \leq \infty $, the quotient map $C^{\ast }_{L^{p+}}(G) \twoheadrightarrow C^{\ast }_{L^{q+}}(G)$ is not injective. We prove that whenever $G$, moreover, acts transitively on the vertices of $T$ and has Tits’s independence property, the group $C^{\ast }$-algebras $C^{\ast }_{L^{p+}}(G)$ are the only group $C^{\ast }$-algebras of $G$ coming from ideals of the Fourier–Stieltjes algebra. We also show that given such a group $G$, every group $C^{\ast }$-algebra $C^{\ast }_{\mu }(G)$ that is distinguishable from $C^{\ast }(G)$ and whose dual space $C^{\ast }_{\mu }(G)^{\ast }$ is an ideal in $B(G)$ is abstractly $^{\ast}$-isomorphic to $C^{\ast }_{r}(G)$.