Convolution powers in the operator-valued framework
HTML articles powered by AMS MathViewer
- 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.
References
- Michael Anshelevich, Free evolution on algebras with two states, J. Reine Angew. Math. 638 (2010), 75–101. MR 2595336, DOI 10.1515/CRELLE.2010.003
- Serban T. Belinschi and Alexandru Nica, $\eta$-series and a Boolean Bercovici-Pata bijection for bounded $k$-tuples, Adv. Math. 217 (2008), no. 1, 1–41. MR 2357321, DOI 10.1016/j.aim.2007.06.015
- Serban T. Belinschi and Alexandru Nica, On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution, Indiana Univ. Math. J. 57 (2008), no. 4, 1679–1713. MR 2440877, DOI 10.1512/iumj.2008.57.3285
- Serban T. Belinschi and Alexandru Nica, Free Brownian motion and evolution towards $\boxplus$-infinite divisibility for $k$-tuples, Internat. J. Math. 20 (2009), no. 3, 309–338. MR 2500073, DOI 10.1142/S0129167X09005303
- S. T. Belinschi and H. Bercovici, Atoms and regularity for measures in a partially defined free convolution semigroup, Math. Z. 248 (2004), no. 4, 665–674. MR 2103535, DOI 10.1007/s00209-004-0671-y
- S. T. Belinschi and H. Bercovici, Partially defined semigroups relative to multiplicative free convolution, Int. Math. Res. Not. 2 (2005), 65–101. MR 2128863, DOI 10.1155/IMRN.2005.65
- S. T. Belinschi, M. Popa, and V. Vinnikov, Infinite divisibility and a non-commutative Boolean-to-free Bercovici-Pata bijection, J. Funct. Anal. 262 (2012), no. 1, 94–123. MR 2852257, DOI 10.1016/j.jfa.2011.09.006
- Hari Bercovici and Vittorino Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. (2) 149 (1999), no. 3, 1023–1060. With an appendix by Philippe Biane. MR 1709310, DOI 10.2307/121080
- Stephen Curran, Analytic subordination for free compression, preprint arXiv:0803.4227v2 [math.OA], 2008.
- J. William Helton, Reza Rashidi Far, and Roland Speicher, Operator-valued semicircular elements: solving a quadratic matrix equation with positivity constraints, Int. Math. Res. Not. IMRN 22 (2007), Art. ID rnm086, 15. MR 2376207, DOI 10.1093/imrn/rnm086
- Alexandru Nica and Roland Speicher, On the multiplication of free $N$-tuples of noncommutative random variables, Amer. J. Math. 118 (1996), no. 4, 799–837. MR 1400060
- Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR2266879 (2008k:46198)
- Mihai Popa and Victor Vinnikov, Non-commutative functions and non-commutative free Levy-Hincin formula, arXiv:1007.1932v2 [math.OA], 2010.
- Dimitri Shlyakhtenko, Random Gaussian band matrices and freeness with amalgamation, Internat. Math. Res. Notices 20 (1996), 1013–1025. MR 1422374, DOI 10.1155/S1073792896000633
- Dimitri Shlyakhtenko, $A$-valued semicircular systems, J. Funct. Anal. 166 (1999), no. 1, 1–47. MR 1704661, DOI 10.1006/jfan.1999.3424
- Roland Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627, x+88. MR 1407898, DOI 10.1090/memo/0627
- Roland Speicher and Reza Woroudi, Boolean convolution, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 267–279. MR 1426845
- Dan Voiculescu, Symmetries of some reduced free product $C^\ast$-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588. MR 799593, DOI 10.1007/BFb0074909
- Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. MR 839105, DOI 10.1016/0022-1236(86)90062-5
- Dan Voiculescu, Operations on certain non-commutative operator-valued random variables, Astérisque 232 (1995), 243–275. Recent advances in operator algebras (Orléans, 1992). MR 1372537
- Dan Voiculescu, The coalgebra of the free difference quotient and free probability, Internat. Math. Res. Notices 2 (2000), 79–106. MR 1744647, DOI 10.1155/S1073792800000064
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