Regularity of Polynomials in Free Variables

TL;DR

This paper investigates the spectral measures of non-commutative polynomials in free variables, establishing conditions under which these measures are atomless or non-singular, based on free entropy dimension and additional assumptions.

Contribution

It demonstrates that spectral measures lack atoms when the free entropy dimension equals the number of variables and provides conditions for non-singularity and polynomial decay of measures.

Findings

Spectral measures have no atoms if free entropy dimension equals the number of variables.

Under stronger conditions, spectral measures are not singular.

Measures of intervals around points decay at least polynomially.

Abstract

We show that the spectral measure of any non-commutative polynomial of a non-commutative $n$-tuple cannot have atoms if the free entropy dimension of that $n$-tuple is $n$ (see also work of Mai, Speicher, and Weber). Under stronger assumptions on the $n$-tuple, we prove that the spectral measure is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval's length.