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 .

The original proof of Gleason's Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason's Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason's Theorem, e.g. that on the quantum logic L(H ) of all closed subspaces of a Hilbert space H, dim H ≥ 3, there is no finitely additive state whose range is countably infinite. In particular, if dim H = n, then on L(H ) there is a unique discrete state, namely m(A) = dim A/ dim H, A ∈ L(H ).

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