Vol. 276, No. 2, 2015

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ISSN: 0030-8730
Free evolution on algebras with two states, II

Michael Anshelevich

Vol. 276 (2015), No. 2, 257–280
Abstract

Denote by J the operator of coefficient stripping. We show that for any free convolution semigroup {μt : t 0} with finite variance, applying a single stripping produces semicircular evolution with nonzero initial condition, J [μt] = ρ σβ,γt, where σβ,γ is the semicircular distribution with mean β and variance γ. For more general freely infinitely divisible distributions τ, expressions of the form ρ̃ τt arise from stripping μ̃t, where {(μ̃t,μt) : t 0} 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
Authors
Michael Anshelevich
Department of Mathematics
Texas A&M University
Mailstop 3368
College Station, TX 77843-3368
United States