Hypercontractivity in group von Neumann algebras

Marius Junge, Carlos Palazuelos, Javier Parcet, Mathilde Perrin

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. We will illustrate our method with free groups, triangular groups and finite cyclic groups, for which we shall obtain optimal time hypercontractive L2 → Lq inequalities with respect to the Markov process given by the word length and with q an even integer. Interpolation and differentiation also yield general Lp → Lq hypercontrativity for 1 < p ≤ q < ∞ 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) Lp → Lq 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).

Original languageEnglish (US)
Pages (from-to)1-102
Number of pages102
JournalMemoirs of the American Mathematical Society
Volume249
Issue number1183
DOIs
StatePublished - Sep 2017

Keywords

  • Fourier multiplier
  • Group von Neumann algebra
  • Hypercontractivity

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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