Topological Properties of Operator Generalized Effect Algebras

Abstract

For any dense linear subspace D of a complex Hilbert space H we introduce and study a D-weak operator topology τD on the set G D ( H ) of all positive linear operators with domain D. For instance, we show that every A ∈ G D ( H ) is τD-limit of a nondecreasing sequence of bounded operators from G D ( H ) . Let τw denote the weak operator topology on the set of all bounded linear operators and let B + ( H ) denote the set of all positive bounded linear operators. For the corresponding relative topologies we show that τ D ∩ B + ( H ) ⊂ τ w ∩ B + ( H ) and the converse inclusion need not hold. Some topological properties of the operator generalized effect algebra ( G D ( H ) ; ⊕ , 0 ) and its intervals (all of which are effect algebras) are also studied.