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Finite de Finetti theorems for free easy quantum groups

Jianquan Wang

Abstract

We prove various finite de Finetti theorems for non-commutative distributions which are invariant under the free easy quantum group actions. This complements the free de Finetti theorems by Banica, Curran and Speicher, which mostly focus on infinite sequences. We also discuss some refined results for the infinite setting.

Finite de Finetti theorems for free easy quantum groups

Abstract

We prove various finite de Finetti theorems for non-commutative distributions which are invariant under the free easy quantum group actions. This complements the free de Finetti theorems by Banica, Curran and Speicher, which mostly focus on infinite sequences. We also discuss some refined results for the infinite setting.

Paper Structure

This paper contains 4 sections, 5 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite de Finetti theorem for easy quantum groups
  4. Infinite de Finetti theorem for easy quantum groups

Key Result

Theorem 1

Given $n\geq 4$, let $(x_1, . . . , x_n)$ be a family of random variables in a $\ast$-probability space $(A, \varphi)$. Let $G_n$ be a free easy quantum group with the associated category of non-crossing partitions $\mathcal{C}$. Then the following are equivalent:

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Remark 2
  • Corollary 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • ...and 2 more