Characterization of regular Riesz spaces using oru-compact operators (Q923314)

Let F be an Archimedean Riesz space and, for a positive element x of F, let \(F_ x\) denote the order ideal generated by x. Then \(F_ x\) can be made into a normed linear space by the usual norm given by \(\| y\|_ x=\inf \{\alpha:\alpha >0\), \(| y| \leq \alpha x\}\). A subset M of F is said to be relatively uniformly (ru) totally bounded if, for some positive element x, M is a totally bounded subset of \((F_ x,\| \|_ x)\). A linear operator U on a Riesz space E into an Archimedean Riesz space is said to be oru-compact if U(A) is ru totally bounded in F whenever A is an order bounded subset of E. In an earlier note [C. R. Acad. Sci., Paris Sér. I Math. 303, No.14, 707-710 (1986; Zbl 0611.47031)], the author introduced the space of oru-compact operators as an extension of the space of finite rank operators with good Riesz space properties. The main result of the present paper is the proof of a theorem announced in a previous note [op. cit.]. In order to state the theorem we need additional terms and notations. Let E and F be as above. Then \(L_{oru}(E,F)\) denotes the space of all oru-compact operators on E into F. The space \(L_{oru}(E,F)\) is a Riesz space under the usual ordering if F is ru complete (i.e., \((F_ x,\| \|_ x)\) is complete for each positive x in F), which we assume to be the case below. Let \(L^ x_{oru}(E,F)\) denote the order ideal in \(L_{oru}(E,F)\) consisting of those U for which \(| U|\) is order continuous, and let \(L_ r(E,F)\) denote the space of all order bounded operators on E into F. The space F is said to be super order complete if it is order complete and order separable (i.e., whenever \(M\subset F\) with sup \(M\in F\), there is a countable subset \(N\subset M\) with sup N\(=\sup M)\). The space F is said to be regular if, whenever \(\{x_ n\}\) is a sequence in F order converging to zero, \(\| x_ n\|_ u\to 0\) for some positive u in F. The main theorem states that, if F is a super complete and regular Riesz space, then, for each Riesz space E, \(L_{oru}(E,F)\) and \(L^ x_{oru}(E,F)\) are bands in \(L_ r(E,F)\). Conversely if F is an order complete Riesz space such that \(L_{oru}(\ell_{\infty},F)\) is order closed in \(L_ r(\ell_{\infty},F)\), then F is super order complete and regular. In the last section, the author investigates the relationship between the kernel operators and the oru-compact operators.