Hypercontractivity in Group von Neumann Algebras.

Cover -- Title page -- Introduction -- Chapter 1. The combinatorial method -- 1.1. Notation -- 1.2. Aim of the method -- 1.3. Admissible lengths -- 1.4. Completing squares I -- 1.5. A decomposition of ᵤ( ) -- 1.6. Completing squares II -- 1.7. Analysis of both approaches -- 1.8. Λ-estimates -- 1.9. Δ-estimates -- 1.10. Strategy -- Chapter 2. Optimal time estimates -- 2.1. Free groups -- 2.2. Triangular groups -- 2.3. Finite cyclic groups -- 2.4. Comments -- Chapter 3. Poisson-like lengths -- 3.1. Proof of Theorem B -- 3.2. Behavior of the constant (\G, ) -- 3.3. Examples of Poisson-like lengths -- 3.4. Ultracontractivity -- Appendix A. Logarithmic Sobolev inequalities -- Appendix B. The word length in ℤ_{ } -- Appendix C. Numerical analysis -- C.1. Estimates for free groups -- C.2. Estimates for triangular groups -- C.3. Estimates for finite cyclic groups -- Appendix D. Technical inequalities -- D.0. Positivity test for polynomials -- D.1. Technical inequalities for free groups -- D.2. Technical inequalities for triangular groups -- D.3. Technical inequalities for finite cyclic groups -- D.4. Proofs -- Bibliography -- Back Cover.

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.

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