Pure Jauch-Piron states on von Neumann algebras

We study Jauch-Piron states and two-valued measures on von Neumann algebra. We prove as the main result that, under some set-theoretical assumption, a pure state of a von Neumann algebra A not containing a central abelian portion is Jauch-Piron if and only if it is σ-additive. Moreover, we show that this result holds for type I factor indenpendently on the set-theoretical axiomatics. As a consequence we obtain a lucid characterization of pure Jauch-Piron states on von Neumann algebras acting on a Hilbert space with real-nonmeasurable dimension (this can be viewed as a generalization of the paper). We also characterize the von Neumann algebras whose logic of projections is Jauch-Piron. Finally, we prove that every two-valued measure on the projection logic of A, where A contains no type I2 central portion, has to be concentrated at an abelian direct summand of A. (orig.)