The order topology for a von Neumann algebra

Abstract

The order topology τo(P) (resp. the sequential order topology τos(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 Msa, the self-adjoint part of the unit ball M1sa, 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 operator-algebraic structure of M.