Unification of Independence in Quantum Probability

Abstract

Let [Formula: see text], be the conditionally free product of unital free *-algebras [Formula: see text], where ϕl, ψl are states on [Formula: see text], l∈I. We construct a sequence of noncommutative probability spaces [Formula: see text], m∈N, where [Formula: see text] and [Formula: see text], m∈N, [Formula: see text], and the states [Formula: see text], ϕl are Boolean extensions of ϕl, ψl, l∈I, respectively. We define unital *-homomorphisms [Formula: see text] such that [Formula: see text] converges pointwise to *l∈I(ϕl,ψl). Thus, the variables j(m)(w), where w is a word in [Formula: see text], converge in law to the conditionally free variables. The sequence of noncommutative probability spaces [Formula: see text], where [Formula: see text] and Φ(m) is the restriction of [Formula: see text] to [Formula: see text], is called a hierarchy of freeness. Since all finite joint correlations for known examples of independence can be obtained from tensor products of appropriate *-algebras, this approach can be viewed as a unification of independence. Finally, we show how to make the m-fold free product [Formula: see text] into a cocommutative *-bialgebra associated with m-freeness.