  The order topology $\tau_o(P)$ (resp. the sequential order topology
$\tau_{os}(P)$) 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 $M$.
