Abstract

The projection latticesP(ℳ1),P(ℳ2) of two von Neumann subalgebras ℳ1, ℳ2 of the von Neumann algebra ℳ are defined to be logically independent if A ∧ B≠0 for any 0≠AεP(ℳ1), 0≠BP(ℳ2). After motivating this notion in independence, it is shown thatP(ℳ1),P(ℳ2) are logically independent if ℳ1 is a subfactor in a finite factor ℳ andP(ℳ1),P(ℳ2 commute. Also, logical independence is related to the statistical independence conditions called C*-independence W*-independence, and strict locality. Logical independence ofP(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, thenP(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