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
Msa1​, 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