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