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

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 Aut(T ) of a (semi-)homogeneous tree T that acts transitively on the boundary ∂T and every 2 ≤ q < p ≤ ∞, the canonical quotient mapThis 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 * µ (G) that is distinguishable (as a group C * -algebra) from the universal group C * -algebra of G and whose dual space C * µ (G) * is a G-invariant ideal in B(G) is abstractly * -isomorphic to the reduced group C * -algebra of G.