Unification of independence in quantum probability

Let ((*l is an element of I)A, (*l is an element of I)(phi(l), psi(l))), be the conditionally free product of unital free *-algebras A(l), where phi(l), psi(l) are states on A(l), l is an element of I. We construct a sequence of noncommutative probability spaces ((A) over tilde((m)), <(Phi)over tilde>((m))), m is an element of N, where (A) over tilde(m) = X-l is an element of I (A) over tilde(l)(Xm) and <(Phi)over tilde>((m)) = X-l is an element of I <(phi)over tilde>l X <(psi)over tilde>(X(m-1))(l), m is an element of N, (A) over tilde(l) = A * C[t], and the states <(phi)over tilde>l, <(psi)over tilde>(l) are Boolean extensions of phi(l), psi(l), l is an element of I, respectively. We define unital *-homomorphisms j((m)) : (*l is an element of I) A(l) --> (A) over tilde((m)) such that <(Phi)over tilde>((m)) circle j((m)) converges pointwise to (*l is an element of I)(phi(l), psi(l)). Thus, the variables j((m))(w), where w is a word in (*l is an element of I) A(l), converge in law to the conditionally free variables. The sequence of noncommutative probability spaces (A((m)), Phi((m))), where A((m)) = j((m)) ((*l is an element of I) A(l)) and Phi((m)) is the restriction of <(Phi)over tilde>((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 (A) over tilde*((m)) into a cocommutative *-bialgebra associated with m-freeness.