An operator characterization of continuous Riesz spaces

AbstractWe show that the orthogonality of order bounded finite rank operators T:E→E to the identity operator on E is equivalent to the continuity of the space E. We also describe discrete elements in the space Lb(E,F) of order bounded linear maps transforming a Riesz space E into a Dedekind complete Riesz space F. Our description is the same as in Wickstead (1981) [5] but we obtain it making less restrictive, more natural assumptions and presenting a different proof. Additionally, we formulate a necessary and sufficient condition for the discreteness and continuity of Lb(E,F)