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Freely independent random variables with non-atomic distributions


Authors: Dimitri Shlyakhtenko and Paul Skoufranis
Journal: Trans. Amer. Math. Soc. 367 (2015), 6267-6291
MSC (2010): Primary 46L54; Secondary 15B52
Published electronically: February 26, 2015
MathSciNet review: 3356937
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Abstract: We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups, thus proving that the measure of each atom of any $ n \times n$ matricial polynomial of non-atomic, freely independent random variables is an integer multiple of $ n^{-1}$. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic, and thus the polynomial has a distribution that is real-analytic except at a finite number of points.


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Additional Information

Dimitri Shlyakhtenko
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
Email: shlyakht@math.ucla.edu

Paul Skoufranis
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
Email: pskoufra@math.ucla.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06434-4
Keywords: Freely independent random variables, non-atomic distributions, Atiyah Property for tracial $*$-algebras, free entropy, semicircular variables.
Received by editor(s): June 12, 2013
Published electronically: February 26, 2015
Additional Notes: This research was supported in part by NSF grants DMS-090076, DMS-1161411, DARPA HR0011-12-1-0009, and by NSERC PGS
Article copyright: © Copyright 2015 American Mathematical Society