Statistical independence of operator algebras

In the paper we investigate statistical independence of C*-algebras and its relation to other independence conditions studied in operator algebras and quantum field theory. Especially, we prove that C*-algebras A(1) and A(2) are statistically independent if and only if for every normalized elements a is an element of A(1) and b is an element of A(2) there is a state phi of the whole algebra such that phi(a) = phi(b) = 1. As a consequence we show that logical independence (see [17, 18]) implies statistical independence and that statistical independence implies independence in the sense of Schlieder. We prove that the reverse implications are not valid, Further, independence of commuting algebras is shown to be equivalent to independence of their centers. Finally, results on independence of commuting algebras are generalized to the context of Jordan-Banach algebras.