## 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:

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