Free evolution on algebras with two states II

Abstract

Denote by $J$ the operator of coefficient stripping. We show that for any free convolution semigroup of measures $\nu_t$ with finite variance, applying a single stripping produces semicircular evolution with non-zero initial condition, $J[\nu_t] = \rho \boxplus \sigma^{\boxplus t}$, where $\sigma$ is the semicircular distribution with mean $\beta$ and variance $\gamma$. For more general freely infinitely divisible distributions $\tau$, expressions of the form $\rho \boxplus \tau^{\boxplus t}$ arise from stripping $\mu_t$, where the pairs $(\mu_t, \nu_t)$ form 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.