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A Characterization of the Leinert property

Authors:	Franz Lehner

Institution:	Institut für Mathematik, Johannes Kepler Universität Linz, A4040 Linz, Austria

Abstract:	Let be a discrete group and denote by $\lambda _G$ its left regular representation on $\ell _2(G)$ . Denote further by ${\mathbf {F}}_n$ the free group on generators $\{g_1,g_2,\ldots ,g_n\}$ and $\lambda$ its left regular representation. In this paper we show that a subset $S=\{ t_1, t_2, \ldots , t_n \}$ of has the Leinert property if and only if for some real positive coefficients $\alpha _1,\alpha _2,\ldots ,\alpha _n$ the identity $\begin{displaymath}\biggl \\| \sum _{i=1}^n \alpha _i \, \lambda _G(t_i) \biggr \\|_{C_\lambda ^(G)} = \biggl \\| \sum _{i=1}^n \alpha _i \, \lambda (g_i) \biggr \\|_{C_\lambda ^({\mathbf {F}}_n)} \end{displaymath}$ holds. Using the same method we obtain some metric estimates about abstract unitaries $U_1,U_2,\ldots , U_n$ satisfying the similar identity $\biggl \\|\sum _{i=1}^n U_i \otimes \overline {U_i}\biggr \\|_{\min }$ $=2\sqrt {n-1}.$

Keywords:	Norm of a convolution operator Leinert property free group random walk

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