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

Belinschi, S. T.; Popa, M.; Vinnikov, V.

J. Operator Theory 70 (2013), no. 1, 239–258

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.