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A new representation of the Dedekind completion of $C(K)$-spaces


Authors: Z. Ercan and S. Onal
Journal: Proc. Amer. Math. Soc. 133 (2005), 3317-3321
MSC (2000): Primary 46A40; Secondary 46B42, 54B42
DOI: https://doi.org/10.1090/S0002-9939-05-07889-5
Published electronically: May 9, 2005
MathSciNet review: 2161155
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Abstract: A new representation of the Dedekind completion of $C(K)$ is given. We present a necessary and sufficient condition on a compact Hausdorff space $K$ for which the Dedekind completion of $C(K)$ is $B(S)$, the space of real valued bounded functions on some set $S$.


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  • 1. C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz spaces with Applications to Economics, $2^{nd}$ Edition, Mathematical Surveys and Monographs, Volume 105, American Mathematical Society, Providence, RI, 2003. MR 2011364
  • 2. R. P. Dilworth, The normal completion of the lattice of continuous functions, Trans. Amer. Math. Soc. 68 (1950), 427-438. MR 0034822 (11:647g)
  • 3. W. Filter, Wolfgang Representations of Archimedean Riesz spaces-a survey, Rocky Mountain J. Math. 24 (1994), no. 3, 771-851.MR 1307578 (96f:46006)
  • 4. S. Kaplan, The Bidual of $C(X)$ I, North-Holland, New York and Amsterdam, 1985. MR 0776606 (86k:46001)
  • 5. S. Kaplan, Lebesgue Theory in the Bidual of $C(X)$, Memoirs Amer. Mat. Society 121 (1996), no. 579. MR 1329941 (96i:46023)
  • 6. W. A. J. Luxemburg and A.C. Zaanen, Riesz spaces I, North-Holland Publ. Comp., Amsterdam, 1971. MR 0511676 (58:23483)
  • 7. J. E. Mack and D.G. Johnson, The Dedekind completion of $C(X)$, Pacific J. Math. 20 (1967), 231-243. MR 0211268 (35:2150)
  • 8. W. Maxey, The Dedekind completion of $C(X)$ and its second dual, Ph.D. Thesis (1973), Purdue Univ.
  • 9. N. Nakano, Über das System aller stetigen Funktionen auf einen Topologischen Raum, Proc. Imp. Acad. (Tokyo) 17 (1941), 308-310.MR 0014173 (7:249f)
  • 10. N. Nakano and T. Shimogaki, A note on the cut extension of C-spaces, Proc. Japan. Acad. 38 (1962), 473-477. MR 0149254 (26:6744)
  • 11. A. I. Veksler, Functional representation of the order complement of a vector lattice of continuous functions, Sibirsk. Mat. Zh. 26 (1985), no. 6, 159-162. MR 0816514 (87j:46048a)
  • 12. E. C. Weinberg, The alpha-completion of ring of continuous real-valued functions, Notices Amer. Math. Soc. 7 (1960), 533-534.
  • 13. V. K. Zaharov, Functional characterization of absolute and Dedekind completion, Bull. Acad. Polon. Sci. Ser. Math. 29, no.5-6 (1981), 293-297. MR 0640475 (83b:54019)
  • 14. V. K. Zaharov, Functional characterization of the absolute, vector lattices of functions with Baire property and of quasinormal functions and modules of particular continuous functions, Trudy Moskov. Mat. Obshch. 45 (1982), 68-104. MR 0704628 (85c:46033)

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Additional Information

Z. Ercan
Affiliation: Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Email: zercan@metu.edu.tr

S. Onal
Affiliation: Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Email: osul@metu.edu.tr

DOI: https://doi.org/10.1090/S0002-9939-05-07889-5
Keywords: Riesz spaces, Dedekind completion
Received by editor(s): November 30, 2003
Received by editor(s) in revised form: June 21, 2004
Published electronically: May 9, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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