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Sums of Commutators in Free Probability

Wiktor Ejsmont, Franz Lehner

Abstract

We study the linear span of commutators of free random variables and show that these are the only quadratic forms which satisfy the following equivalent properties: * preservation free infinite divisibility * free and strong cancellation of odd cumulants * symmetric distribution for any free family. The main combinatorial tool is an involution on non-crossing partitions.

Sums of Commutators in Free Probability

Abstract

We study the linear span of commutators of free random variables and show that these are the only quadratic forms which satisfy the following equivalent properties: * preservation free infinite divisibility * free and strong cancellation of odd cumulants * symmetric distribution for any free family. The main combinatorial tool is an involution on non-crossing partitions.

Paper Structure

This paper contains 25 sections, 17 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notation and Terminology
  4. Free Independence
  5. Free Convolution and the Cauchy-Stieltjes Transform
  6. Free infinite divisibility
  7. Some probability distributions
  8. Wigner semicircle law
  9. Free Poisson law
  10. Tetilla law
  11. Compound free Poisson distribution
  12. Even elements
  13. Noncrossing Partitions
  14. Free Cumulants
  15. Cumulants of quadratic forms
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $X_1, X_2,\dots,X_n\in\mathcal{A}$ be random variables in a tracial probability space, then

Figures (3)

  • Figure 3: Two adjacent inner odd blocks.
  • Figure 4: Examples of flip partitions
  • Figure 10: Examples of the involution of partitions of type III.

Theorems & Definitions (40)

  • Lemma 2.1
  • Theorem 2.2
  • Definition 2.3: NicaSpeicher:2006
  • Lemma 2.4: EjsmontLehner:2017
  • Proposition 2.5: EjsmontLehner:2017
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 30 more