Unimodality for free multiplicative convolution with free normal distributions on the unit circle

We study unimodality for free multiplicative convolution with free normal distributions λtt>0 on the unit circle. We give four results on unimodality for μ⊠λt: (1) if μ is a symmetric unimodal distribution on the unit circle then so is μ⊠λt at any time t>0; (2) if μ is a symmetric distribution on T supported on eiθ: θ∈−φ, φ for some φ∈ (0, π2), then μ⊠λt is unimodal for sufficiently large t>0; (3) b⊠λt is not unimodal at any time t>0, where b is the equally weighted Bernoulli distribution on 1, −1; (4) λt is not freely strongly unimodal for sufficiently small t>0. Moreover, we study unimodality for classical multiplicative convolution, which is useful in proving the above four results.