Hypercontractivity in Group von Neumann Algebras
Seiten
2017
American Mathematical Society (Verlag)
978-1-4704-2565-4 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-2565-4 (ISBN)
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Provides a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. The authors illustrate their method with free groups, triangular groups and finite cyclic groups, for which they 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.
In this paper, the authors provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. They illustrate their method with free groups, triangular groups and finite cyclic groups, for which they 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. The authors' 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 on a computer.
The authors 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. The authors' 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).
In this paper, the authors provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. They illustrate their method with free groups, triangular groups and finite cyclic groups, for which they 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. The authors' 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 on a computer.
The authors 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. The authors' 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).
Marius Junge, University of Illinois at Urbana-Champaign, Illinois. Carlos Palazuelos, Instituto de Ciencias Matematicas, Madrid, Spain. Javier Parcet, Instituto de Ciencias Matematicas, Madrid, Spain. Mathilde Perrin, Instituto de Ciencias Matematicas, Madrid, Spain.
The combinatorial method
Optimal time estimates
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
Bibliography.
| Erscheinungsdatum | 12.10.2017 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 160 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| ISBN-10 | 1-4704-2565-3 / 1470425653 |
| ISBN-13 | 978-1-4704-2565-4 / 9781470425654 |
| Zustand | Neuware |
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Buch | Softcover (2022)
Springer Spektrum (Verlag)
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