How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

Hypercontractivity in group von Neumann algebras


About this Title

Marius Junge, Carlos Palazuelos, Javier Parcet and Mathilde Perrin

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

View full volume PDF

View other years and numbers:

Table of Contents


Chapters

  • Introduction
  • Chapter 1. The combinatorial method
  • Chapter 2. Optimal time estimates
  • Chapter 3. Poisson-like lengths
  • Appendix A. Logarithmic Sobolev inequalities
  • Appendix B. The word length in $\mathbb Z_n$
  • Appendix C. Numerical analysis
  • Appendix D. Technical inequalities

Abstract


In this paper, we provide a combinatorial/numerical method to establish -1pt new -1pt hypercontractivity estimates in group -1pt von Neumann algebras. -3pt We will illustrate our method with free groups, triangular groups and finite cyclic groups, for which we shall obtain optimal time hypercontractive inequalities with respect to the Markov process given by the word length and with an even integer. Interpolation and differentiation also yield general hypercontrativity for 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) 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).

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