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

TL;DR

This paper explores the operator-valued analogues of classical probability laws such as semicircle, arcsine, and Bernoulli, revealing their relations under different convolutions and providing combinatorial and analytical characterizations.

Contribution

It demonstrates that operator-valued distributions satisfy scalar-like relations only in the fully matricial setting and offers a combinatorial and analytical framework for these distributions.

Findings

Operator-valued distributions follow scalar relations in the fully matricial sense.

A combinatorial description of the operator-valued arcsine distribution is provided.

The reciprocal Cauchy transform of the operator-valued arcsine satisfies a version of the Abel equation.

Abstract

We study of 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 measures 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.