Orthomorphisms on Riesz spaces of Riesz space-valued functions. (English) Zbl 0774.46010

Lavrič, Boris

Math. Balk., New Ser. 5, No. 1, 15-19 (1991).

Summary: Let \(E\) be an Archimedean Riesz space, \(X\) a nonempty set, and \({\mathcal F}(X,E)\) the Riesz space of all functions \(X\to E\). Suppose that \(L\) is a Riesz subspace of \({\mathcal F}(X,E)\) satisfying \(\{f(x)\); \(f\in L\}=E\) for all \(x\in X\). It is shown that the \(f\)-algebra \(\text{Orth}(L)\) can be embedded via a function \(\Phi\) into the \(f\)-algebra \({\mathcal F}(X,\text{Orth}(E))\), and that \(\Phi\) embeds the center \(Z(L)\) with its uniform norm isometrically into \({\mathcal F}_ b(X,Z(E))\) equipped with the supremum norm. If \(E\) is a Banach lattice, \(X\) a locally compact Hausdorff topological space, and \(L\) a Riesz subspace of \({\mathcal C}(X,E)\) satisfying some additional conditions, then \(\Phi\) maps \(\text{Orth}(L)\) (resp. \(Z(L)\)) onto the \(f\)-algebra \({\mathcal C}^ s(X,Z(E))\) (resp. \({\mathcal C}_ b^ s(X,Z(E))\)) of all strongly continuous (and bounded) \(Z(E)\)- valued functions on \(X\).

MSC:

46A40	Ordered topological linear spaces, vector lattices
46B42	Banach lattices

Keywords:

Archimedean Riesz space; \(f\)-algebra; Banach lattice