Free evolution on algebras with two states, II

Michael Anshelevich

Vol. 276 (2015), No. 2, 257–280

DOI: 10.2140/pjm.2015.276.257

Abstract

Denote by $\mathcal{J}$ 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, $\mathcal{J}\left[{\mu }_t\right]=\rho ⊞{\sigma }_{\beta ,\gamma }^{⊞t}$, 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 $\tilde{\rho }⊞{\tau }^{⊞t}$ arise from stripping ${\tilde{\mu }}_t$, where $\left\{\left({\tilde{\mu }}_t,{\mu }_t\right):t\ge \mathfrak{0}\right\}$ 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

Mathematical Subject Classification 2010

Primary: 46L54

Milestones

Received: 1 April 2012

Revised: 24 June 2014

Accepted: 19 December 2014

Published: 15 July 2015