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

Szwarc, Ryszard

J. Funct. Anal. 95, No. 2, 320-343 (1991).

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 \(A(G)\).

Reviewer: E.Płonka (Wodzisław)

MSC:

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

Keywords:

Hilbert space; irreducibility; unitary representations; regular representations; semihomogeneous tree; Fourier algebra; approximate unit; multiplier norm

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