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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A vector lattice topology and function space representation

Authors: W. A. Feldman and J. F. Porter
Journal: Trans. Amer. Math. Soc. 235 (1978), 193-204
MSC: Primary 46E05; Secondary 46A40, 54C40
MathSciNet review: 0463897
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Abstract: A locally convex topology is defined for a vector lattice having a weak order unit and a certain partition of the weak order unit, analogous to the order unit topology. For such spaces, called β€œorder partition spaces,” an extension of the classical Kakutani theorem is obtained: Each order partition space is lattice isomorphic and homeomorphic to a dense subspace of ${C_c}(X)$ containing the constant functions for some locally compact X, and conversely each such ${C_c}(X)$ is an order partition space. $({C_c}(X)$ denotes all continuous real-valued functions on X with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of ${C_c}(X)$ for X locally compact and realcompact. Conditions for an M-space to be an order partition space are provided.

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Keywords: Compact convergence, function space, topological <I>M</I>-space, vector lattice, weak order unit
Article copyright: © Copyright 1978 American Mathematical Society