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