The order topology for a von Neumann algebra

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 $M_{sa}^1$, 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 $M$.

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