Radu groups acting on trees are CCR (Q6622309)

A locally compact group \(G\) is called CCR if the operator \(\pi(f)\) is compact for all irreducible representations \(\pi\) of \(G\) and all \(f \in L^{1}(G)\). For totally disconnected locally compact groups, this property is equivalent to the requirement that every irreducible representation of \(G\) is admissible [\textit{C. Nebbia}, Rocky Mt. J. Math. 29, No. 1, 311--316 (1999; Zbl 0931.43008)].\N\NIn this paper, the author classifies the irreducible unitary representations of closed simple groups of automorphisms of trees acting \(2\)-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, he confirms the CCR conjecture by Nebbia [loc. cit.] on trees for \((d_{0},d_{1})\)-semi-regular trees such that \(d_{0}, d_{1}\in \Theta\) where \(\Theta\) is an asymptotically dense set of positive integers.