A vector lattice topology and function space representation

Feldman, W. A.; Porter, J. F.

doi:10.1090/S0002-9947-1978-0463897-8

A vector lattice topology and function space representation
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by W. A. Feldman and J. F. Porter PDF

Trans. Amer. Math. Soc. 235 (1978), 193-204 Request permission

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 “order 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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