Topological Properties of Operator Generalized Effect Algebras

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