On freely indecomposable measures

TL;DR

This paper characterizes when a probability measure cannot be decomposed into nontrivial free convolutions, using analytic subordination techniques for both additive and multiplicative cases.

Contribution

It provides new criteria for free indecomposability of measures, extending previous results to both additive and multiplicative free convolutions.

Findings

Measures with no mass in an interval between atoms are indecomposable

Results apply to both free additive and multiplicative convolutions

Analytic subordination is used as the main proof technique

Abstract

We show that a probability measure is not a nontrivial free additive convolution if it puts no mass in an interval whose endpoints are atoms. The analogous results for free multiplicative convolutions are proved as well. The proofs use analytic subordination.