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 -algebra . We show how for a -valued distribution one can define convolution powers (with respect to free additive convolution) and (with respect to Boolean convolution), where the exponent is a suitably chosen linear map from to , instead of being a nonnegative real number. More precisely, is always defined when is completely positive, while is always defined when is completely positive (with ``'' denoting the identity map on ).

In connection to these convolution powers we define an evolution semigroup , completely positive, related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the -valued free Brownian motion.

We also obtain two results on the operator-valued analytic function theory related to convolution powers . One of the results concerns the analytic subordination of the Cauchy-Stieltjes transform of with respect to the Cauchy-Stieltjes transform of . The other one gives a -valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a -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

Email:
manshel@math.tamu.edu

**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

Email:
belinsch@math.usask.ca

**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

Email:
fevrier@math.univ-toulouse.fr, maxime.fevrier@math.u-psud.fr

**Alexandru Nica**

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

Email:
anica@math.uwaterloo.ca

DOI:
http://dx.doi.org/10.1090/S0002-9947-2012-05736-9

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.