Free evolution on algebras with two states II

TL;DR

This paper studies how a single coefficient stripping operation transforms free convolution semigroups of measures, revealing semicircular evolution and providing formulas for generators, with numerous examples and algebraic insights.

Contribution

It demonstrates that coefficient stripping produces semicircular evolution in free convolution semigroups and characterizes the structure of these transformations, including algebraic and explicit examples.

Findings

Stripping yields semicircular evolution with non-zero initial conditions.

Expressions of the form ρ ⊞ τ^{⊞ t} arise from stripping in free convolution semigroups.

Provides formulas for generators of these semigroups.

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.