Continuity and growth of free multiplicative convolution semigroups

Xiaoxue Deng,Ping Zhong

doi:10.1214/18-ecp202

Abstract

Let $\mu $ be a compactly supported probability measure on the positive half-line and let $\mu ^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $\mu ^{\boxtimes t}$ varies continuously as $t$ changes. We also obtain the asymptotic length of the support of these measures.

Highlights

Let μ and ν be probability measures on [0, ∞)
We will prove the following result about continuity of the support of μ⊠t, which is the analogue of the work [16] for free multiplicative convolution on the positive half line
We study the asymptotic size of the support

Summary

Introduction

Let μ and ν be probability measures on [0, ∞). The free convolution μ ⊠ ν represents the distribution of product of two positive operators in a tracial W ∗-probability space whose distributions are μ and ν respectively. We will prove the following result about continuity of the support of μ⊠t, which is the analogue of the work [16] for free multiplicative convolution on the positive half line. The free multiplicative convolution on the unit circle is usually studied together with the positive half line case. It was shown in [1] that many results can be deduced from results on free additive convolutions. Since analogue results were known in additive case, a separate work on free multiplicative convolution on the unit circle become unnecessary.

Free convolution on the positive half line

Continuity of free convolution semigroups

Estimation of norm of free multiplicative convolution semigroups