We classify 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, we confirm Claudio Nebbia’s CCR conjecture 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.

Let $G$ be a locally compact group and $\pi $ a representation of $G$ by weakly* continuous isometries acting in a dual Banach space $E$ . Given a probability measure $\mu $ on $G$ , we study the Choquet–Deny equation $\pi (\mu )x\,=\,x,\,x\,\in \,E$ . We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $\mu $ . The relation between the space of solutions of the Choquet–Deny equation in $E$ and the space of bounded harmonic functions can be understood in terms of a construction resembling the ${{W}^{*}}$ -crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space $E\,=\,B({{L}^{2}}(G))$ of bounded linear operators on ${{L}^{2}}(G)$ . Other general properties of the Choquet–Deny equation in a Banach space are also discussed.

Let $G$ be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra $L^{1}_{\omega}(G)$ is a symmetric Banach $*$-algebra if and only if the weight function $\omega $ satisfies the GRS-condition. This condition expresses in a precise technical sense that $\omega $ grows subexponentially.

Let $G$ be a compact group.

Let $\sigma$ be a continuous involution of $G$ . In this paper, we are concerned by the following functional equation

$$\int\limits_{G}{f\left( xty{{t}^{-1}} \right)dt}\,+\int\limits_{G}{f\left( xt\sigma \left( y \right){{t}^{-1}} \right)dt}=2g\left( x \right)h\left( y \right),\,\,\,x,y\in G,$$

where $f,g,h:G\mapsto \mathbb{C}$ , to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d’Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$ .

It is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an Lp -space, for 1 < p < ∞ p ≠ 2, can be dilated by a strongly continuous group of (sub)positive isometries in a manner analogous to the dilation M. A. Akçoglu and L. Sucheston constructed for a discrete semigroup of (sub)positive contractions. From this an improvement of a von Neumann type estimation, due to R. R.Coifman and G.Weiss, on the transfer map belonging to the semigroup is deduced.