#### Vol. 276, No. 2, 2015

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Free evolution on algebras with two states, II

### Michael Anshelevich

Vol. 276 (2015), No. 2, 257–280
##### Abstract

Denote by $\mathsc{J}$ the operator of coefficient stripping. We show that for any free convolution semigroup $\left\{{\mu }_{t}:t\ge \mathfrak{0}\right\}$ with finite variance, applying a single stripping produces semicircular evolution with nonzero initial condition, $\mathsc{J}\left[{\mu }_{t}\right]=\rho ⊞{\sigma }_{\beta ,\gamma }^{⊞t}$, where ${\sigma }_{\beta ,\gamma }$ is the semicircular distribution with mean $\beta$ and variance $\gamma$. For more general freely infinitely divisible distributions $\tau$, expressions of the form $\stackrel{̃}{\rho }⊞{\tau }^{⊞t}$ arise from stripping ${\stackrel{̃}{\mu }}_{t}$, where $\left\{\left({\stackrel{̃}{\mu }}_{t},{\mu }_{t}\right):t\ge \mathfrak{0}\right\}$ forms a semigroup under the operation of two-state free convolution. The converse to this statement holds in the algebraic setting. Numerous examples illustrating these constructions are computed. Additional results include the formula for generators of such semigroups.

##### Keywords
free convolution semigroups, two-state free convolution, coefficient stripping, subordination distribution
Primary: 46L54