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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 \(A(G)\).

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
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References:

[1] Bouaziz, F, ()
[2] Bożejko, M; Fendler, G, Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. un. mat. ital. D (6), 3-A, 297-302, (1984) · Zbl 0564.43004
[3] {\scM. Bożejko and M. Picardello}, paper in preparation.
[4] Cowling, M; Haagerup, U, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, (1987), School of Math., University New South Wales, preprint
[5] de Canniere, J; Haagerup, U, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. math. J., 107, 455-500, (1984) · Zbl 0577.43002
[6] Julg, P; Valette, A, K-moyennabilité pour LES groupes opérant sur un arbre, C. R. acad. sci. Paris Sér. I math., 296, 977-980, (1983) · Zbl 0537.46055
[7] Julg, P; Valette, A, K-amenability for SL(2,\(Q\)p), and the action on the associated tree, J. funct. anal., 58, 194-215, (1984) · Zbl 0559.46030
[8] Ol’shanskii, G.I, Classification of irreducible representations of groups of automorphisms of Bruhat-Tits trees, Functional anal. appl., 11, No. 1, 26-34, (1977) · Zbl 0371.22014
[9] Pytlik, T; Szwarc, R, An analytic family of uniformly bounded representations of free groups, Acta math., 157, 287-309, (1986) · Zbl 0681.43011
[10] Serre, J.P, Arbres, amalgames, SL2, Astérisque, 46, (1977)
[11] Szwarc, R, An analytic series of irreducible representations of the free group, Ann. inst. Fourier (Grenoble), 38, 87-110, (1988) · Zbl 0634.22003
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