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Completeness type properties of locally solid Riesz spaces. (English) Zbl 0543.46002
Let \(\Omega\) be an extremally disconnected topological space, \(C^{\infty}(\Omega)\) the Riesz space of continuous functions from \(\Omega\) into \({\mathbb{R}}^{\infty}\) which take finite values on dense subsets of \(\Omega\), where \({\mathbb{R}}^{\infty}\) denotes the real line compactified by \(+\infty\) and -\(\infty\). Function filters and topological vector cores are introduced and studied in \(C^{\infty}(\Omega)\). These are in parallel with the function norm and Banach function spaces in the Luxemburg-Zaanen theory. Armed with this generalization, the following main results are proved.
Let \((L,\tau)\) be a Hausdorff topological Riesz space (the terminologies are as in C. D. Aliprantis and O. Burkinshaw’s book ’Locally Solid Riesz Spaces’ (1978; Zbl 0402.46005). \((L,\tau)\) embeds order densely into a Nakano space \((L^{\#},\tau^{\#})\) if and only if \(\tau\) is Fatou; this embedding is unique up to an isomorphism. A Dedekind complete \((L,\tau)\) embeds order densely into a Hausdorff locally solid Dedekind complete Riesz space \((L^{\#},\tau^{\#})\) having the monotone completeness property if and only if \(\tau\) is pseudo-Lebesgue.
Reviewer: Ng Kung-Fu

46A40 Ordered topological linear spaces, vector lattices
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