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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A characterization of the Leinert property
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by Franz Lehner PDF
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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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
  • Email: lehner@caddo.bayou.uni-linz.ac.at, lehner@imada.ou.dk
  • Received by editor(s): February 22, 1996
  • Received by editor(s) in revised form: May 21, 1996
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3423-3431
  • MSC (1991): Primary 22D25; Secondary 43A05, 43A15, 60J15
  • DOI: https://doi.org/10.1090/S0002-9939-97-03966-X
  • MathSciNet review: 1402870