States on projection logics of von Neumann algebras

Andrzej Łuczak

Jan Hamhalter

Jan Hamhalter

Carlo Cecchini

Etang Chen

The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*-algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n < + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.

Yuichiro Kitajima

In algebraic quantum field theory, some von Neumann algebra is associated with any finite open set in Minkowski space-time. Such a von Neumann algebra is called a local algebra. Let V1 and V2 be two spacelikely separated spacetime regions and let N(V1) and N(V2) be local algebras associated with V1 and V2 respectively. In the present paper it is shown that N(V1) and N(V2) are logically independent if and only if any correlation between projections of N(V1) and N(V2) has a Reichenbach's pre-common cause explanation.