Lecture Notes on "Non-Commutative Distributions"

TL;DR

This paper introduces the theory of non-commutative distributions, covering free analysis, operator-valued free probability, and rational functions, aiming to provide a comprehensive, accessible overview for readers without prior free probability knowledge.

Contribution

It extends free probability theory into a more general, operator-valued framework, integrating analytic and algebraic methods for non-commutative distributions.

Findings

Develops free analysis for non-commuting variables

Introduces operator-valued free probability theory

Explains the linearization trick for polynomials in free variables

Abstract

This in an introduction to the theory of non-commutative distributions of non-commuting operators or random matrices. Starting from the basic problem to find a good approach to the meaning of "non-commutative distribution" we will, in particular, cover: free analysis, which is a version of complex analysis for several non-commuting variables; the operator-valued version of free probability theory (combinatorial but also analytic aspects); the linearization trick to reduce non-linear scalar problems to linear operator-valued problems; the combination of operator-valued convolution and linearization to calculate the distribution of polynomials in free variables; the basic theory of non-commutative rational functions. On one hand, this is a continuation of the Lecture Notes on "Free Probability", arXiv:1908.08125. On the other hand, the theory of free probability is developed again, but in a more general, operator-valued context. So, in principle and with some additional efforts, it should be possible to read the present notes without having a prior knowledge on free probability. Big parts of the material do also not deal so much with free variables, but more general with analytic and algebraic aspects of maximal non-commuting variables. The material here was presented in the summer term 2019 at Saarland University in 20 lectures of 90 minutes each. The lectures were recorded and can be found online at https://www.math.uni-sb.de/ag/speicher/web_video/index.html.