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 | References | Similar Articles | Additional Information

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 containing the constant functions for some locally compact *X*, and conversely each such is an order partition space. denotes all continuous real-valued functions on *X* with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of for *X* locally compact and realcompact. Conditions for an *M*-space to be an order partition space are provided.

**[1]**Frank W. Anderson,*Approximation in systems of real-valued continuous functions*, Trans. Amer. Math. Soc.**103**(1962), 249–271. MR**0136976**, 10.1090/S0002-9947-1962-0136976-0**[2]**E. Binz,*Notes on a characterization of function algebras*, Math. Ann.**186**(1970), 314–326. MR**0264397****[3]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[4]**W. A. Feldman and J. F. Porter,*Compact convergence and the order bidual for 𝐶(𝑋)*, Pacific J. Math.**57**(1975), no. 1, 113–124. MR**0374878****[5]**W. A. Feldman and J. F. Porter,*Order units and base norms generalized for convex spaces*, Proc. London Math. Soc. (3)**33**(1976), no. 2, 299–312. MR**0417740****[6]**D. H. Fremlin,*Topological Riesz spaces and measure theory*, Cambridge University Press, London-New York, 1974. MR**0454575****[7]**M. Henriksen and D. G. Johnson,*On the structure of a class of archimedean lattice-ordered algebras.*, Fund. Math.**50**(1961/1962), 73–94. MR**0133698****[8]**Edwin Hewitt,*Linear functionals on spaces of continuous functions*, Fund. Math.**37**(1950), 161–189. MR**0042684****[9]**Graham J. O. Jameson,*Topological 𝑀-spaces*, Math. Z.**103**(1968), 139–150. MR**0222605****[10]**Shizuo Kakutani,*Concrete representation of abstract (𝑀)-spaces. (A characterization of the space of continuous functions.)*, Ann. of Math. (2)**42**(1941), 994–1024. MR**0005778****[11]**R. G. Kuller,*Locally convex topological vector lattices and their representations.*, Michigan Math. J.**5**(1958), 83–90. MR**0097706****[12]**Anthony L. Peressini,*Ordered topological vector spaces*, Harper & Row, Publishers, New York-London, 1967. MR**0227731****[13]**Donald Plank,*Closed 𝑙-ideals in a class of lattice-ordered algebras*, Illinois J. Math.**15**(1971), 515–524. MR**0280423****[14]**Claude Portenier,*Espaces de Riesz, espaces de fonctions et espaces de sections*, Comment. Math. Helv.**46**(1971), 289–313 (French). MR**0291764**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0463897-8

Keywords:
Compact convergence,
function space,
topological *M*-space,
vector lattice,
weak order unit

Article copyright:
© Copyright 1978
American Mathematical Society