Free-Boolean independence for pairs of algebras

Abstract

We construct pairs of algebras with mixed independence relations by using truncations of reduced free products of algebras. For example, we construct free-Boolean pairs of algebras and free-monotone pairs of algebras. We also introduce free-Boolean cumulants and show that free-Boolean independence is equivalent to the vanishing of mixed cumulants.

Free-Boolean independence with amalgamation

“…In this section, we review some combinatorial tools which will be used to define operator-valued free-Boolean cumulants. We give a characterization of free-Boolean independence with amalgamation thereby generalizes results in [8] to the operator-valued framework. In noncommutative probability theory, non-crossing partitions are used in the combinatorics of free probability and the interval partitions are used in the combinatorics of Boolean independence.…”

“…In noncommutative probability theory, non-crossing partitions are used in the combinatorics of free probability and the interval partitions are used in the combinatorics of Boolean independence. It turns out the partitions used in the combinatorics of free-Boolean independence are so-called interval-noncrossing partitions introduced in [8]. All results without proof in this section are taken from [8].…”

“…It turns out the partitions used in the combinatorics of free-Boolean independence are so-called interval-noncrossing partitions introduced in [8]. All results without proof in this section are taken from [8].…”

“…there are exactly two unital universal independence relations, namely Voiculescu's free independence relation, the classical independence relation [15]. It was mentioned that we would obtain more independence relations by decreasing the number of axioms for universal products [8]. For instance, people introduced Boolean independence [17], monotone independence [10], conditionally independence [1] in various contexts.…”

“…In [8], the first-named author introduced a notion of mixed independence relations for pairs of random variables, where random variables in different faces exhibit different kinds of noncommutative independence. In particular, the combinatorics of free-Boolean independence relation was fully developed.…”