Free evolution on algebras with two states II

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

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