Groups acting on trees and approximation properties of the Fourier algebra (Q1174984)

Let \( Aut X\) be the group of all isometries of a tree \(X\) (=connected graph without circuits). A series of representations parametrized by \(z\in\mathbb{C}\), \(| z|<1\) in the Hilbert space \(l^ 2(X)\) is constructed. The author investigates the irreducibility, the question of existence of equivalent unitary representations and the connections with regular representations in the case \(X\) being a semihomogeneous tree \(X_{a,b}\) (in any vertex meet \(a+1\) or \(b+1\) edges and the vertices of any edge have different degrees). The following result is proved: For any group \(G\) acting on a tree in such a way that the stabilizer of a vertex is a compact subgroup of \(G\) the Fourier algebra \(A(G)\) admits an approximate unit bounded in the multiplier norm on \(A(G)\).