On Regularity for Measures in Multiplicative Free Convolution Semigroups

Abstract

Given a probability measure μ on the real line, there exists a semigroup μ t with real parameter t > 1 which interpolates the discrete semigroup of measures μ n obtained by iterating its free convolution. It was shown in (Math. Z. 248(4):665–674, 2004) that it is impossible that μ t have 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.