A characterization of the Leinert property

Lehner, Franz

doi:10.1090/S0002-9939-97-03966-X

A characterization of the Leinert property
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Proc. Amer. Math. Soc. 125 (1997), 3423-3431 Request permission

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 \[ \biggl \| \sum _{i=1}^n \alpha _i \lambda _G(t_i) \biggr \|_{C_\lambda ^*(G)} = \biggl \| \sum _{i=1}^n \alpha _i \lambda (g_i) \biggr \|_{C_\lambda ^*(\mathbf {F}_n)} \] 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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