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

Ercan, Z.; Onal, S.

doi:10.1090/S0002-9939-05-07889-5

A new representation of the Dedekind completion of $C(K)$-spaces
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by Z. Ercan and S. Onal PDF

Proc. Amer. Math. Soc. 133 (2005), 3317-3321 Request permission

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 $S$.

References

Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, 2nd ed., Mathematical Surveys and Monographs, vol. 105, American Mathematical Society, Providence, RI, 2003. MR 2011364, DOI 10.1090/surv/105
R. P. Dilworth, The normal completion of the lattice of continuous functions, Trans. Amer. Math. Soc. 68 (1950), 427–438. MR 34822, DOI 10.1090/S0002-9947-1950-0034822-9
Wolfgang Filter, Representations of Archimedean Riesz spaces—a survey, Rocky Mountain J. Math. 24 (1994), no. 3, 771–851. MR 1307578, DOI 10.1216/rmjm/1181072375
Samuel Kaplan, The bidual of $C(X)$. I, North-Holland Mathematics Studies, vol. 101, North-Holland Publishing Co., Amsterdam, 1985. MR 776606
Samuel Kaplan, Lebesgue theory in the bidual of $C(X)$, Mem. Amer. Math. Soc. 121 (1996), no. 579, viii+127. MR 1329941, DOI 10.1090/memo/0579
W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1971. MR 0511676
J. E. Mack and D. G. Johnson, The Dedekind completion of $C({\scr X})$, Pacific J. Math. 20 (1967), 231–243. MR 211268, DOI 10.2140/pjm.1967.20.231
W. Maxey, The Dedekind completion of $C(X)$ and its second dual, Ph.D. Thesis (1973), Purdue Univ.
Hidegorô Nakano, Über das System aller stetigen Funktionen auf einem topologischen Raum, Proc. Imp. Acad. Tokyo 17 (1941), 308–310 (German). MR 14173
Kazumi Nakano and Tetsuya Shimogaki, A note on the cut extension of $C$-spaces, Proc. Japan Acad. 38 (1962), 473–477. MR 149254
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, 191 (Russian). MR 816514
E. C. Weinberg, The alpha-completion of ring of continuous real-valued functions, Notices Amer. Math. Soc. 7 (1960), 533-534.
Valerij Zaharov, Functional characterization of absolute and Dedekind completion, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), no. 5-6, 293–297 (English, with Russian summary). MR 640475
V. K. Zakharov, Functional characterization of the absolute vector lattices of functions with the Baire property and of quasinormal functions, and modules of quotients of continuous functions, Trudy Moskov. Mat. Obshch. 45 (1982), 68–104 (Russian). MR 704628