Generalized Effect Algebras of Bounded Positive Operators Defined on Hilbert Spaces

It is shown that compact positive operators, Hilbert–Schmidt positive operators and trace-class positive operators form sub-generalized effect algebras of the generalized effect algebra of positive bounded operators on Hilbert space (with the ⊕ operation being the usual operator sum) and they are generalized effect algebras in their own right. The intersections of these sets with the set of Hilbert space effects form three nontrivial sub-generalized effect algebras of the generalized effect algebra of Hilbert space effects. Also a more general version of these results is given.