LOGICAL INDEPENDENCE IN QUANTUM LOGIC

The projection lattices P(M1), P(M2) of two von Neumann subalgebras M1, M2 of the von Neumann algebra M are defined to be logically independent if A AND B not-equal 0 for any 0 not-equal A is-an-element-of P(M1), 0 not-equal B is-an-element-of P(M2). After motivating this notion in independence, it is shown that P(M1), P(M2) are logically independent if M1 is a subfactor in a finite factor M and P(M1), P(M2) commute. Also, logical independence is related to the statistical independence conditions called C*-independence W*-independence, and strict locality. Logical independence of P(M1), P(M2) turns out to be equivalent to the C*-independence of (M1, M2) for mutually commuting M1, M2, and it is shown that if (M1, M2) is a pair of (not necessarily commuting) von Neumann subalgebras, then P(M1), P(M2) are logically independent in the following cases: M is a finite-dimensional full-matrix algebra and M1, M2 are C*-independent; (M1, M2) is a W*-independent pair; M1, M2 have the property of strict locality.