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Heat semigroups on quantum automorphism groups of finite dimensional C*-algebras

Futaba Sato

TL;DR

This work analyzes heat semigroups on the quantum automorphism group Aut^+(B) equipped with the Plancherel trace. It proves ultracontractivity, hypercontractivity, log-Sobolev inequalities, and a spectral gap, with explicit constants tied to the dimension of B. It also establishes the sharp Sobolev embedding and Hausdorff-Young inequalities for Aut^+(B) (dim B ≥ 5), leveraging monoidal equivalence to S_n^+ and a Drinfeld double/tube-algebra perspective for alternative proofs. The results extend known properties from quantum permutation groups to the broader class Aut^+(B) and provide precise, operator-norm estimates for the heat semigroups in this noncommutative setting.

Abstract

In this paper, we investigate heat semigroups on a quantum automorphism group ${\rm Aut}^+(B)$ of a finite dimensional C*-algebra $B$ and its Plancherel trace. We show ultracontractivity, hypercontractivity, and the spectral gap inequality of the heat semigroups on ${\rm Aut}^+(B)$. Furthermore, we obtain the sharpness of the Sobolev embedding property and the Hausdorff-Young inequality of ${\rm Aut}^+(B)$.

Heat semigroups on quantum automorphism groups of finite dimensional C*-algebras

TL;DR

This work analyzes heat semigroups on the quantum automorphism group Aut^+(B) equipped with the Plancherel trace. It proves ultracontractivity, hypercontractivity, log-Sobolev inequalities, and a spectral gap, with explicit constants tied to the dimension of B. It also establishes the sharp Sobolev embedding and Hausdorff-Young inequalities for Aut^+(B) (dim B ≥ 5), leveraging monoidal equivalence to S_n^+ and a Drinfeld double/tube-algebra perspective for alternative proofs. The results extend known properties from quantum permutation groups to the broader class Aut^+(B) and provide precise, operator-norm estimates for the heat semigroups in this noncommutative setting.

Abstract

In this paper, we investigate heat semigroups on a quantum automorphism group of a finite dimensional C*-algebra and its Plancherel trace. We show ultracontractivity, hypercontractivity, and the spectral gap inequality of the heat semigroups on . Furthermore, we obtain the sharpness of the Sobolev embedding property and the Hausdorff-Young inequality of .

Paper Structure

This paper contains 14 sections, 21 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of compact quantum groups
  4. Quantum automorphism groups of finite dimensional C*-algebras
  5. Heat semigroups on ${\rm Aut}^+(B)$
  6. Noncommutative $L^p$ spaces
  7. Ultracontracitivity and hypercontractivity of heat semigroups on ${\rm Aut}^+(B)$
  8. Ultracontractivity
  9. Hypercontractivity and the log-Sobolev inequality
  10. Spectral gap inequality
  11. The sharp Sobolev embedding properties for ${\rm Aut}^+(B)$
  12. The sharp Sobolev embedding properties
  13. The sharp Hausdorff-Young inequality
  14. Appendix: another proof for the formula of heat semigroups on ${\rm Aut}^+(B)$

Key Result

Theorem 3

The set of classes of finite dimensional irreducible representations of ${\rm Aut}^+(B, \psi)$ with a $\delta$-form $\psi$ can be labeled by the positive integers, ${\rm Irr}({\rm Aut}^+(B, \psi))=\{U_k:k\in\mathbb{N}\}$. The fundamental representation $U$ sarisfies $U\cong 1\oplus U_1$ and the fusi

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Example 4
  • Example 5
  • Proposition 6
  • Theorem 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 32 more