The order topology for a von Neumann algebra

Abstract

Abstract. The order topology τ o (P ) (resp. the sequential order topology τ os (P )) on a poset P is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra M we consider the following three posets: the self-adjoint part M sa , the self-adjoint part of the unit ball M 1 sa , and the projection lattice P (M ). We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other standard locally convex topologies on M , and relate the properties of the order topology to the underlying operatoralgebraic structure of M .

Vigier's theorem for the spectral order and its applications

“…is true only for abelian von Neumann algebras [6], we see that τ o (M sa , ) and τ o (M sa , ≤) coincide if and only if M is abelian.…”

The order topology on duals of $C^\ast $-algebras and von Neumann algebras

“…Then 0 ≤ xn i ≤ yi ≤ 1. Let us verify that the sequence (yi) i∈N is decreasing (see also Lemma 5.2 in [11]). Indeed,…”

“…We mention, for example, the beautiful result of Floyd and Klee [8] saying that a normed space X is reflexive if and only if the order topology on the lattice of all closed subspaces of X ordered by set-theoretic inclusion is Hausdorff. Continuing this line of research, the order topology on the lattice of closed subspaces of a Hilbert space was studied in [16,11]. A systematic treatment of various aspects of order topologies on structures associated with von Neumann algebras was carried out in [11].…”

“…Continuing this line of research, the order topology on the lattice of closed subspaces of a Hilbert space was studied in [16,11]. A systematic treatment of various aspects of order topologies on structures associated with von Neumann algebras was carried out in [11]. In this case the order topology comes from the standard operator order defined on the self-adjoint operators acting on a Hilbert space H; namely, for self-adjoint operators T and S, we have T ≤ S if the operator S − T has a positive spectrum.…”

“…In this case the order topology comes from the standard operator order defined on the self-adjoint operators acting on a Hilbert space H; namely, for self-adjoint operators T and S, we have T ≤ S if the operator S − T has a positive spectrum. The work in [11] showed that the order topology on the self-adjoint part of a von Neumann algebra, albeit being far from being a linear topology, on bounded parts it coincides -in a surprising way -with the strong operator topology. Besides, finite von Neumann algebras were characterized in terms of the order topology on its projection lattice.…”