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Hypercontractivity in group von Neumann algebras
About this Title
Marius Junge, Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St. Urbana, Illinois, Carlos Palazuelos, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, C/ Nicolás Cabrera 13-15. Madrid. Spain, Javier Parcet, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, C/ Nicolás Cabrera 13-15. Madrid. Spain and Mathilde Perrin, Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, C/ Nicolás Cabrera 13-15. Madrid. Spain
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 249, Number 1183
ISBNs: 978-1-4704-2565-4 (print); 978-1-4704-4133-3 (online)
DOI: https://doi.org/10.1090/memo/1183
Published electronically: August 8, 2017
Keywords: Hypercontractivity,
Fourier multiplier,
group von Neumann algebra
MSC: Primary 22D15, 43A22, 47D07
Table of Contents
Chapters
- Introduction
- 1. The combinatorial method
- 2. Optimal time estimates
- 3. Poisson-like lengths
- A. Logarithmic Sobolev inequalities
- B. The word length in $\mathbb {Z}_n$
- C. Numerical analysis
- D. Technical inequalities
Abstract
In this paper, we provide a combinatorial/numerical method to establish \hskip-1pt new \hskip-1pt hypercontractivity estimates in group \hskip-1pt von Neumann algebras. \hskip-3pt We will illustrate our method with free groups, triangular groups and finite cyclic groups, for which we shall obtain optimal time hypercontractive $L_2 \to L_q$ inequalities with respect to the Markov process given by the word length and with $q$ an even integer. Interpolation and differentiation also yield general $L_p \to L_q$ hypercontrativity for $1 < p \le q < \infty$ via logarithmic Sobolev inequalities. Our method admits further applications to other discrete groups without small loops as far as the numerical part (which varies from one group to another) is implemented and tested in a computer. We also develop another combinatorial method which does not rely on computational estimates and provides (non-optimal) $L_p \to L_q$ hypercontractive inequalities for a larger class of groups/lengths, including any finitely generated group equipped with a conditionally negative word length, like infinite Coxeter groups. Our second method also yields hypercontractivity bounds for groups admitting a finite dimensional proper cocycle. Hypercontractivity fails for conditionally negative lengths in groups satisfying Kazhdan’s property (T).- Mats Erik Andersson, Remarks on hypercontractivity for the smallest groups, Complex analysis and differential equations (Uppsala, 1997) Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., vol. 64, Uppsala Univ., Uppsala, 1999, pp. 51–60. MR 1758914
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