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On the Haagerup inequality and groups acting on \(\tilde A_ n\)-buildings. (English) Zbl 0886.51003
Summary: Let \(\Gamma\) be a group endowed with a length function \(L\), and let \(E\) be a linear subspace of \(\mathbb{C}\Gamma\). We say that \(E\) satisfies the Haagerup inequality if there exist constants \(C,s>0\) such that, for any \(f\in E\), the convolutor norm of \(f\) on \(\ell^{2}(\Gamma)\) is dominated by \(C\) times the \(\ell^{2}\) norm of \(f(1+L)^{s}\). We show that, for \(E=\mathbb{C}\Gamma\), the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on \(\Gamma\). If \(L\) is a word length function on a finitely generated group \(\Gamma\), we show that, if the space \(\text{Rad}_{L}(\Gamma)\) of radial functions with respect to \(L\) satisfies the Haagerup inequality, then \(\Gamma\) is non-amenable if and only if \(\Gamma\) has superpolynomial growth. We also show that the Haagerup inequality for \(\text{Rad}_{L}(\Gamma)\) has a purely combinatorial interpretation; thus, using the main result of the companion paper by J. Swiatkowski, we deduce that, for a group \(\Gamma\) acting simply transitively on the vertices of a thick euclidean building of type \(\tilde{A}_{n}\), the space \(\text{Rad}_{L}(\Gamma)\) satisfies the Haagerup inequality, and \(\Gamma\) is non-amenable.

51E24 Buildings and the geometry of diagrams
60G50 Sums of independent random variables; random walks
44A35 Convolution as an integral transform
43A05 Measures on groups and semigroups, etc.
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