On regularity for measures in multiplicative free convolution semigroups

Abstract

Given a probability measure $\mu$ on the real line, there exists a semigroup ${\mu }_t$ with real parameter $t>1$ which interpolates the discrete semigroup of measures ${\mu }_n$ obtained by iterating its free convolution. It was shown in \cite{[BB2004]} that it is impossible that ${\mu }_t$ has no mass in an interval whose endpoints are atoms. We extend this result to semigroups related to multiplicative free convolution. The proofs use subordination results.

Some typos fixed, accepted by Complex Analysis and Operator Theory