States and Structure of von Neumann Algebras

We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory.     

On the de Morgan Property of the Standard Brouwer–Zadeh Poset

The standard Brouwer–Zadeh poset (H) is the poset of all effect operators on a Hilbert space H, naturally equipped with two types of orthocomplementation. In developing the theory, the question occured if (when) (H) fulfils the de Morgan property with respect to both orthocomplementation operations. In Ref.3 the authors proved that it is the case provided dimH<, and they conjectured that if dimH=, then the answer is in the negative. In this note, we first give a somewhat simpler proof of the known result for dimH<, and then we give a proof to the conjecture: We show that if dimH=, then the de Morgan property is not valid.     

VITALI-HAHN-SAKS THEOREM FOR VECTOR MEASURES ON OPERATOR ALGEBRAS

We show that the direct generalization of the Vitali–Hahn–Sakstheorem is not valid for all measures on von Neumann algebras.By applying a general equicontinuity argument, we prove a directextension of the Vitali–Hahn–Saks theorem for awide range of vector measures on von Neumann algebra s and JBWalgebras. We also characterize relatively compact sets of vectormeasures on operator algebras.     

Extension properties of states on operator algebras

We summarize and deepen some recent results concerning the extension problem for states on operator algebras and general quantum logics. In particular, we establish equivalence between the Gleason extension property, the Hahn-Banach extension property, and the universal state extension property of projection logics. Extensions of Jauch-Piron states are investigated.     

Absolute Continuity in Noncommutative Measure Theory

Recent results on absolute continuity of Banach space valued operators and convergence theorems on operator algebras are deepened and summarized. It is shown that absolute continuity of an operator T on a von Neumann algebra M with respect to a positive normal functional ψ on M is not implied by the fact that the null projections of ψ are the null projections of T. However, it is proved that the implication above is true whenever M is finite or T is weak^*-continuous. Further it is shown that the absolute value preserves the Vitali-Hahn-Saks property if, and only if, the underlying algebra is finite. This result improves classical results on weak compactness of sets of noncommutative measures.     

Affiliated subspaces and infiniteness of von Neumann algebras

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.     

Determinacy of States and Independence of Operator Algebras

The aim of this paper is to summarize, deepen,and comment upon recent results concerning states onoperator algebras and their extensions. The first partis focused on the relationship between pure states and singly generated subalgebras. Among otherswe show that every pure state on a separablealgebra A is uniquely determined by some element of Awhich exposes . The main part of this paper is the second section, dealing with characterizationof various types of independence conditions arising inthe axiomatics of quantum field theory. These twotopics, seemingly different, are connected by a common extension technique based on determinacy ofpure states.     

States and Structure of von Neumann Algebras

We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory.     

Additivity of vector gleason measures

We study the degree of additivity of orthogonal Hilbert-space-valued measures on the latticeL(H) of all projections acting on a Hilbert spaceH. We present criteria for such measures to be completely additive and we establish the connection between the additivity of orthogonal measures and the size of almost disjoint families on dimH. [For example, we show that everyH-valued orthogonal measure is weakly-additive iff (dimH) > dim H]. As a corollary we see that finitely additive orthogonal measures distinguish dimensions of Hilbert spaces (this can be viewed as a generalization of a theorem by Kruszynski). As a further corollary, we obtain that, for cardinals, with >,3, there is no Jordan homomorphism from a typeI -factor into a typeI -factor. Finally, we show that every latticeL(H) with (dimH) = dimH admits a nonzero free orthogonal measure with values inH. Our results contribute to the noncommutative probability theory and also may find applications in the theory of the representation ofC ^*-algebras.