Groups acting on trees and approximation properties of the Fourier algebra. (English) Zbl 0744.22005

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 \(A(G)\).


22D12 Other representations of locally compact groups
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
20E08 Groups acting on trees
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