Canonical random variables for multivariate, algebra-valued distributions

TL;DR

This paper develops a framework for constructing canonical random variables in algebra-valued noncommutative probability spaces, using B-valued cumulants and Fock space representations, with conditions for traciality.

Contribution

It introduces a reformulation of Speicher's B-valued cumulants and constructs canonical variables for arbitrary B-valued random variables in a unified algebraic setting.

Findings

Constructed canonical B-valued random variables on Fock space.

Provided conditions for the traciality of B-valued traces.

Extended the algebraic theory of noncommutative probability with new tools.

Abstract

In the algebraic theory of algebra-valued noncommutative probability spaces, for a unital algebra B, a mild reformulation of Speicher's noncrossing B-valued cumulants for random variables in these spaces is used to construct canonical random variables, acting on a Fock space over B, for arbitrary families of B-valued random variables. Also, a condition for traciality of a trace on B composed with the B-valued conditional expectation is given in terms of B-valued cumulants.