The order topology on the projection lattice of a Hilbert space

Let L(H) denote the complete lattice of projections on a Hilbert space H. On L(H), besides the restriction of the norm and the strong operator topologies (denoted by τu and τs, respectively) one can consider the order topology τo. In Palko (1995) [10] the topologies τo, τs and τu are compared and it is asked whether τs=τu∩τo. Apart from answering this question, showing that τs and τu∩τo are in general different, this paper contributes to the further understanding of the order topology τo and its relation with τs and τu. It is shown that if H is separable and B is a block, i.e. a maximal Boolean sublattice, of L(H), then the restrictions of τs and τu∩τo to B are equal. We also show if (Pi) is a sequence of compact projections, then Pi→0 w.r.t. τs if and only if Pi→0 w.r.t. τo.