@MISC{Rédei95logicalindependence, author = {Miklós Rédei}, title = {Logical Independence in Quantum Logic}, year = {1995} }

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Abstract

The projection lattices P(M 1 ); P(M 2 ) of two von Neumann subalgebras M 1 ; M 2 of the von Neumann algebra M are defined to be logically independent if A ^ B 6= 0 for any 0 6= A 2 P(M 1 ); 0 6= B 2 P(M 2 ). After motivating this notion of independence it is shown that P(M 1 ); P(M 2 ) are logically independent if M 1 is a subfactor in a finite factor M and P(M 1 ); P(M 2 ) commute. Also, logical independence is related to the statistical independence conditions called C -independence W - independence and strict locality. Logical independence of P(M 1 ); P(M 2 ) turns out to be equivalent to the C - independence of (M 1 ; M 2 ) for mutually commuting M 1 ; M 2 , and it is shown that if (M 1 ; M 2 ) is a pair of (not necessarily commuting) von Neumann subalgebras, then P(M 1 ); P(M 2 ) are logically independent if (M 1 ; M 2 ) is a W -independent pair or if M 1 ; M 2 have the property of strict locality.