Kakutani–Krein compact space of the CDw(X)-space in terms of X×{0,1}

AbstractThe unital AM-spaces (AM-spaces with strong order unit) CDw(X) 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 CDw(X) in terms of X×{0,1}. 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 A⊂iso(X) and B⊂iso(Y) such that X∖A and Y∖B are homeomorphic whenever CDw(X) and CDw(Y) 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 f:X→X such that f(U)ΔU is at most countable for each clopen set U and {x:f(x)≠x} 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