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A Characterization of the Leinert property

Author: Franz Lehner
Journal: Proc. Amer. Math. Soc. 125 (1997), 3423-3431
MSC (1991): Primary 22D25; Secondary 43A05, 43A15, 60J15
MathSciNet review: 1402870
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Abstract: Let $G$ 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 $n$ 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 $G$ 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 $\biggl \|\sum _{i=1}^n U_i \otimes \overline {U_i}\biggr \|_{\min }$
$=2\sqrt {n-1}.$

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Additional Information

Franz Lehner
Affiliation: Institut für Mathematik, Johannes Kepler Universität Linz, A4040 Linz, Austria
Address at time of publication: IMADA, Odense Universitet, Campusvej 55, DK 5230 Odense M, Denmark

Keywords: Norm of a convolution operator, Leinert property, free group, random walk
Received by editor(s): February 22, 1996
Received by editor(s) in revised form: May 21, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society