A vector lattice representation theorem and a characterization of locally compact Hausdorff spaces
Certain normed vector lattices were called by \textit{S. Kakutani} abstract (M)-spaces [Ann. Math. II. Ser. 42, 994-1024 (1941; Zbl 0060.266)]. He showed that an (M)-space with unit element is isomorphic to a function space C(X), where X is compact Hausdorff. The author gives representations for what he calls M-partition spaces. These are vector lattices V having an M-partition P, which is a set of positive elements of V closed under the join operation, and satisfying certain conditions. He shows that every M-partition space is a topological M-space in the sense of \textit{G. J. O. Jameson} [Math. Z. 103, 139-150 (1968; Zbl 0173.153)]. The author proves extensions of the Tykhonov theorem on the compactness of product spaces, and of the Stone-Weierstrass theorem. He represents M- partition spaces as dense subspaces of the space (C(Y),k) of continuous real functions on locally compact Hausdorff space Y, with the compact- open topology. He shows that a Tykhonov space Y is a locally compact Hausdorff space iff (C(Y),k) is an M-partition space. The space Y is compact if the M-partition P is finite.
scientific article; zbMATH DE number 3953542