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A concrete description of $CD_{0}(K)$-spaces as $C(X)$-spaces and its applications


Author: Z. Ercan
Journal: Proc. Amer. Math. Soc. 132 (2004), 1761-1763
MSC (2000): Primary 46E05
DOI: https://doi.org/10.1090/S0002-9939-03-07235-6
Published electronically: October 29, 2003
MathSciNet review: 2051138
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Abstract: We prove that for a compact Hausdorff space $K$ without isolated points, $CD_{0}(K)$and $C(K\times \{0,1\})$ are isometrically Riesz isomorphic spaces under a certain topology on $K\times \{0,1\}$. Moreover, $K$ is a closed subspace of $K\times \{0,1\}$. This provides concrete examples of compact Hausdorff spaces $X$ such that the Dedekind completion of $C(X)$ is $B(S)$ (= the set of all bounded real-valued functions on $S$) since the Dedekind completion of $CD_{0}(K)$ is $B(K)$ ( $CD_{0}(K,E)$ and $CD_{w}(K,E)$ spaces as Banach lattices).


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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-03-07235-6
Received by editor(s): October 21, 2002
Received by editor(s) in revised form: January 16, 2003, and February 11, 2003
Published electronically: October 29, 2003
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society

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