The order topology on the projection lattice of a Hilbert space

Abstract

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.