LOGICALLY INDEPENDENT VON-NEUMANN LATTICES

Three definitions of logical independence of two von Neumann lattices P(M(1)), P(M(2)) Of two sub-von Neumann algebras M(1), M(2) of a von Neumann algebra M are given and the relations of the definitions clarified. It is shown that under weak assumptions the following notion, called ''logical independence'' is the strongest: A boolean AND B not equal 0 for any 0 not equal A epsilon P(M(1)), 0 not equal B epsilon P(M(2)). Propositions relating logical independence of P(M(1)), P(M(2)) to C*-independence, W*-independence, and strict locality of M(1), M(2) are presented.