Unification of Independence in Quantum Probability

Let (∗l∈IA, ∗l∈I(φl, ψl)), be the conditionally free product of unital free *algebras Al, where φl, ψl are states on Al, l ∈ I. We construct a sequence of noncommutative probability spaces (Ã(m), Φ̃(m)), m ∈ N, where Ã(m) = ⊗ l∈I Ã ⊗m l and Φ̃ (m) = ⊗ l∈I φ̃l ⊗ ψ̃ ⊗(m−1) l , m ∈ N, Ãl = A ∗ C[t], and the states φ̃l, ψ̃l are Boolean extensions of φl, ψl, l ∈ I, respectively. We define unital *-homomorphisms j(m) : ∗l∈IAl → Ã (m) such that Φ̃(m) ◦ j(m) converges pointwise to ∗l∈I(φl, ψl). Thus, the variables j (m)(w), where w is a word in ∗l∈IAl, converge in law to the conditionally free variables. The sequence of noncommutative probability spaces (A(m),Φ(m)), where A(m) = j(∗l∈IAl) and Φ (m) is the restriction of Φ̃(m) to A(m), 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 Ã∗(m) into a cocommutative *-bialgebra associated with m-freeness.