Some properties of D-weak operator topology

Abstract

Abstract Let 𝓖 D (𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖 D (𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖 D (𝓗) is the whole 𝓖 D (𝓗). The closure of the set of all unbounded elements of 𝓖 D (𝓗) is also the set 𝓖 D (𝓗). If Q is arbitrary unbounded element of 𝓖 D (𝓗), it determines an interval in 𝓖 D (𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.