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Kakutani–Krein compact space of the -space in terms of
- Source :
-
Journal of Mathematical Analysis & Applications . Jan2006, Vol. 313 Issue 2, p611-631. 21p. - Publication Year :
- 2006
-
Abstract
- Abstract: The unital AM-spaces (AM-spaces with strong order unit) are introduced and studied in [Y.A. Abramovich, A.W. Wickstead, Remarkable classes of unital AM-spaces, J. Math. Anal. Appl. 180 (1993) 398–411] for quasi-Stonean spaces X without isolated points. The isometries between these spaces are studied in [Y.A. Abramovich, A.W. Wickstead, A Banach–Stone theorem for new class of Banach spaces, Indiana Univ. Math. J. 45 (1996) 709–720]. In this paper for a compact Hausdorff space X we give a description of the Kakutani–Krein compact Hausdorff space of in terms of . This construction is motivated from the Alexandroff Duplicate of X, which we employ to give a description of the isometries between these spaces. Under some certain conditions we show that for given compact Hausdorff spaces X and Y there exist finite sets and such that and are homeomorphic whenever and are isometric. This is a generalization of one of the main results of [Y.A. Abramovich, A.W. Wickstead, A Banach–Stone theorem for new class of Banach spaces, Indiana Univ. Math. J. 45 (1996) 709–720]. In Example 10 of [Y.A. Abramovich, A.W. Wickstead, A Banach–Stone theorem for new class of Banach spaces, Indiana Univ. Math. J. 45 (1996) 709–720] an infinite quasi-Stonean space has been constructed with some certain properties. We show that the arguments in this example are true for any infinite quasi-Stonean space. In particular, we show that Proposition 11 of [Y.A. Abramovich, A.W. Wickstead, A Banach–Stone theorem for new class of Banach spaces, Indiana Univ. Math. J. 45 (1996) 709–720] is incorrect (but does not affect the main result) of [Y.A. Abramovich, A.W. Wickstead, A Banach–Stone theorem for new class of Banach spaces, Indiana Univ. Math. J. 45 (1996) 709–720]. Finally, we show that for each infinite quasi-Stonean space X there exists a bijection such that is at most countable for each clopen set U and is uncountable. This answers the conjecture in [Y.A. Abramovich, A.W. Wickstead, A Banach–Stone theorem for new class of Banach spaces, Indiana Univ. Math. J. 45 (1996) 709–720] in the negative in a more general setting. [Copyright &y& Elsevier]
- Subjects :
- *COMPLEX variables
*BANACH spaces
*SET theory
*MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 0022247X
- Volume :
- 313
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of Mathematical Analysis & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 19046270
- Full Text :
- https://doi.org/10.1016/j.jmaa.2005.05.063