On the multiplication of operator-valued c-free random variables

TL;DR

This paper explores the multiplication of c-free non-commutative random variables, introducing an analogue of Voiculescu's S-transform to facilitate analysis within this framework.

Contribution

It develops a new S-transform analogue for c-free variables, extending tools for non-commutative probability in operator-valued contexts.

Findings

Constructed an S-transform analogue for c-free variables.

Provided results on the multiplication of c-free non-commutative variables.

Extended the framework of free probability to operator-valued settings.

Abstract

We discuss some results concerning the multiplication of non-commutative random variables that are c-free with respect to a pair $( \Phi, \varphi) $, where $ \Phi $ is a linear map with values in some Banach or C$^\ast$-algebra and $ \varphi $ is scalar-valued. In particular, we construct a suitable analogue of the Voiculescu's $ S $-transform for this framework.