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

It was proved by Samei and Wiersma that for every non-compact, closed subgroup G of the automorphism group Aut(T) of a (semi-)homogeneous tree T acting transitively on the boundary partial derivative T and every 2 <= q < p <= infinity, the quotient map C-Lp+(& lowast;) (G)->> C-Lq+(& lowast;) (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-& lowast;-algebras C-Lp+(& lowast;) (G) are the only group C-& lowast;-algebras of G coming from ideals of the Fourier-Stieltjes algebra. We also show that given such a group G , every group C-& lowast;-algebra C-mu(& lowast;)(G) that is distinguishable from C & lowast;(G) and whose dual space C-mu(& lowast;)(G)(& lowast;) is an ideal in B(G) is abstractly (& lowast;)-isomorphic to C(& lowast;)r (G).