Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $\biggl \|\sum _{i=1}^n U_i \otimes \overline {U_i}\biggr \|_{\min }$
$=2\sqrt {n-1}.$

References [Enhancements On Off] (What's this?)

  • [A-O] Akemann, C.A., Ostrand, P.A., Computing norms in group $C^*$-algebras, Am. J. of Math. 98 (1976) 1015-1047MR 56:1079
  • [B] Bozejko, M., On $\Lambda (p)$ sets with minimal constant in discrete noncommutative groups, Proc. A.M.S. 51 (1975) 407-412 MR 52:11481
  • [C-V] Cherix, A.P., Valette, A., On spectra of simple random walks on one-relator groups, Pac. J. of Math. 175 (1996) 417-438.
  • [H-R-V1] de la Harpe, P., Robertson, A.G., Valette, A., On the Spectrum of the Sum of Generators for a Finitely Generated Group I, Isr. J. of Math. 81 (1993) 65-96 MR 94j:22007
  • [H-R-V] de la Harpe, P., Robertson, A.G., Valette, A., On the spectrum of the sum of generators for a finitely generated group II, Coll. Math. 65 (1993) 87-102
  • [K] Kesten, H., Symmetric Random Walks on Groups, Trans. A.M.S. 92 (1959) 336-354 MR 22:253
  • [Le] Leinert, M., Faltungsoperatoren auf gewissen diskreten Gruppen, Studia math. 52 (1974) 149-158 MR 50:7954
  • [L-P-S] Lubotzky, A., Phillips, R., Sarnak, P., Hecke operators and distributing points on the sphere II, Comm. Pure and Applied Math. 40 (1987) 401-420 MR 88m:11025b
  • [Pa] Paschke, W., Lower bound for the norm of a vertex-transitive graph, Math. Z. 213 (1993) 225-239 MR 94j:05085
  • [P-P] Picardello, M.A., Pytlik, T., Norms of free operators, Proc. A.M.S. 104 (1988) 257-261 MR 90b:47016
  • [P1] Pisier, G., The operator Hilbert space $OH$, complex interpolation and tensor norms, Mem. A.M.S. 122(1996), 103 pgs. MR 97a:46024
  • [P2] Pisier, G., Quadratic forms in unitary operators, to appear
  • [W] Woess, W., A short computation of the norms of free convolution operators, Proc. A.M.S. 96 (1986) 167-170 MR 87e:43002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 22D25, 43A05, 43A15, 60J15

Retrieve articles in all journals with MSC (1991): 22D25, 43A05, 43A15, 60J15

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

American Mathematical Society