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Convolution powers in the operator-valued framework

Authors: Michael Anshelevich, Serban T. Belinschi, Maxime Fevrier and Alexandru Nica
Journal: Trans. Amer. Math. Soc. 365 (2013), 2063-2097
MSC (2010): Primary 46L54
Published electronically: October 4, 2012
MathSciNet review: 3009653
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Abstract: We consider the framework of an operator-valued noncommutative probability space over a unital $ C^*$-algebra $ \mathcal {B}$. We show how for a $ \mathcal {B}$-valued distribution $ \mu $ one can define convolution powers $ \mu ^{\boxplus \eta }$ (with respect to free additive convolution) and $ \mu ^{\uplus \eta }$ (with respect to Boolean convolution), where the exponent $ \eta $ is a suitably chosen linear map from $ \mathcal {B}$ to $ \mathcal {B}$, instead of being a nonnegative real number. More precisely, $ \mu ^{\uplus \eta }$ is always defined when $ \eta $ is completely positive, while $ \mu ^{\boxplus \eta }$ is always defined when $ \eta - 1$ is completely positive (with ``$ 1$'' denoting the identity map on $ \mathcal {B}$).

In connection to these convolution powers we define an evolution semigroup $ \{ \mathbb{B}_{\eta } \mid \eta : \mathcal {B} \to \mathcal {B}$, completely positive$ \}$, related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the $ \mathcal {B}$-valued free Brownian motion.

We also obtain two results on the operator-valued analytic function theory related to convolution powers $ \mu ^{\boxplus \eta }$. One of the results concerns the analytic subordination of the Cauchy-Stieltjes transform of $ \mu ^{\boxplus \eta }$ with respect to the Cauchy-Stieltjes transform of $ \mu $. The other one gives a $ \mathcal {B}$-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a $ \mathcal {B}$-valued free Brownian motion.

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Additional Information

Michael Anshelevich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Serban T. Belinschi
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6 – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania

Maxime Fevrier
Affiliation: Institut de Mathématiques de Toulouse, Equipe de Statistique et Probabilités, F-31062 Toulouse Cedex 09, France
Address at time of publication: Laboratoire de Mathématiques, Université Paris Sud, Bât. 425, 91405 Orsay Cedex, France

Alexandru Nica
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Received by editor(s): August 6, 2011
Published electronically: October 4, 2012
Additional Notes: The first author was supported in part by NSF grant DMS-0900935.
The second author was supported in part by a Discovery Grant from NSERC, Canada, and by a University of Saskatchewan start-up grant.
The third author was supported in part by grant ANR-08-BLAN-0311-03 from Agence Nationale de la Recherche, France.
The fourth author was supported in part by a Discovery Grant from NSERC, Canada.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.