Dual characterization of order continuity and some applications (Q1095380)

If E and F are Riesz spaces, with F Dedekind complete and separated by its order continuous dual \(F^ *_{00}\), then a regular \((=order\) bounded) linear operator \(T: E\to F\) is order continuous iff \(T'(F^ *_{00})\subset E^ *_{00}\). This implies some surprising closure properties of the respective sets of positive and general order continuous operators in L(E,F) and \(L^ r(E,F)\). In fact, those sets are sequentially closed for the weak operator topology with respect to F whenever E is Dedekind \(\sigma\)-complete. If E and F are Banach lattices and F separated by \(F^ *_{00}\), the cone of positive, order continuous operators \(E\to F\) is norm complete.