Regularity of Polynomials in Free Variables
Operator Algebras
2015-08-10 v2
Abstract
We show that the spectral measure of any non-commutative polynomial of a non-commutative -tuple cannot have atoms if the free entropy dimension of that -tuple is (see also work of Mai, Speicher, and Weber). Under stronger assumptions on the -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.
Cite
@article{arxiv.1408.0580,
title = {Regularity of Polynomials in Free Variables},
author = {I. Charlesworth and D. Shlyakhtenko},
journal= {arXiv preprint arXiv:1408.0580},
year = {2015}
}
Comments
The second version (joint with I. Charlesworth) considerably improves our previous results. The main new result is non-singularity of the spectral measure of a non-commutative polynomial of n variables under assumptions of existence of Voiculescu's dual system