On regularity for measures in multiplicative free convolution semigroups

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 \cite{[BB2004]} that it is impossible that $μ_t$ 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.

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