Journal of Operator Theory

Volume 70, Issue 1, Summer 2013 pp. 239-258.

On the operator-valued analogues of the semicircle, arcsine and Bernoulli laws

Authors: S.T. Belinschi (1), M. Popa (2), and V. Vinnikov (3)
Author institution: (1) Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada, and Institute of Mathematics "Simion Stoilow" of the Romanian Academy
(2) Center for Advanced Studies in Mathematics at the Ben Gurion University of the Negev, P.O. B. 653, Be'er Sheva 84105, Israel, and Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-70700, Romania
(3) Department of Mathematics, Ben Gurion University of the Negev, Be'er Sheva 84105, Israel

Summary: We study the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued non-commutative probability these distributions are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that, generally, the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in terms of moments of the operator valued arcsine distribution and we show that its reciprocal Cauchy transform satisfies a version of the Abel equation similar to the one satisfied in the scalar-valued case.

DOI: http://dx.doi.org/10.7900/jot.2011jun24.1963
Keywords: operator-valued distributions, free and Boolean convolutions, generalized Cauchy transform

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