Regularity of Polynomials in Free Variables

2014/08/04 by Ian Charlesworth, Charlesworth, I., Dimitri Shlyakhtenko +1
Mathematics · #46L54 #FOS: Mathematics #Mathematical functions and polynomials #Operator Algebras (math.OA) #Random Matrices and Applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1408.0580

openalex publication_date 2014/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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