Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

Abstract

We show that the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space can be equipped with partial sum of operators making it a generalized effect algebra. This sum coincides with the usual sum of two operators whenever it exists. Moreover, blocks of this generalized effect algebra are proper sub-generalized effect algebras. All intervals in this generalized effect algebra become effect algebras which are Archimedean, convex, interval effect algebras, for which the set of vector states is order determining. Further, these interval operator effect algebras possess faithful states.