On the support of free convolutions

TL;DR

This paper generalizes previous results on the support structure of free convolutions, showing that convolutions of measures with connected supports remain connected, without requiring absolute continuity or boundedness, and extends some results to measures on the unit circle.

Contribution

It extends the connectedness results of free convolutions to arbitrary measures, removing previous restrictions like absolute continuity and bounded support.

Findings

Convolution of measures with connected supports remains connected.

Supports of free additive convolutions can have multiple components, depending on measure properties.

Results are extended to measures on the unit circle.

Abstract

We extend to arbitrary measures results of Bao, Erd\"os, Schnelli, Moreillon, and Ji on the connectedness of the supports of additive convolutions of measures on \mathbb{R} and of free multiplicative convolutions of measures on \mathbb{R}_+. More precisely, the convolution of two measures with connected supports also has connected support. The result holds without any absolute continuity or bounded support hypotheses on the measures being convolved. We also show that the results of Moreillon and Schnelli concerning the number of components of the support of a free additive convolution hold for arbitrary measures with bounded supports. Finally, we provide an approach to the corresponding results in the case of free multiplicative convolutions of probability measures on the unit circle.