Kakutani–Krein compact space of the [formula omitted]-space in terms of [formula omitted

Abstract

The unital AM-spaces ( AM-spaces with strong order unit) CD w ( 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 CD w ( 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 CD w ( X ) and CD w ( 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 ) Δ U 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.