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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Convolution powers in the operator-valued framework
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by Michael Anshelevich, Serban T. Belinschi, Maxime Fevrier and Alexandru Nica PDF
Trans. Amer. Math. Soc. 365 (2013), 2063-2097 Request permission

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
  • 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
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 2063-2097
  • MSC (2010): Primary 46L54
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05736-9
  • MathSciNet review: 3009653