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 τ u and τ s , respectively) one can consider the order topology τ o . In Palko (1995) [10] the topologies τ o , τ s and τ u are compared and it is asked whether τ s = τ u ∩ τ o . Apart from answering this question, showing that τ s and τ u ∩ τ o are in general different, this paper contributes to the further understanding of the order topology τ o and its relation with τ s and τ u . 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 τ s and τ u ∩ τ o to B are equal. We also show if ( P i ) is a sequence of compact projections, then P i → 0 w.r.t. τ s if and only if P i → 0 w.r.t. τ o .