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

Date

2006-01-15

Author

Ercan, Z
Onal, S

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

52
views

0
downloads

The 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) 709-7201. 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 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 subset of iso(X) and B subset of 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) 709720]. 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 I I 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)Delta U is at most countable for each clopen set U and {x: f (x) not equal 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. (c) 2005 Elsevier Inc. All rights reserved.

Subject Keywords

Applied Mathematics, Analysis

URI

https://hdl.handle.net/11511/65671

Journal

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

DOI

https://doi.org/10.1016/j.jmaa.2005.05.063

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

Multinormed semifinite von Neumann algebras, unbounded operators and conditional expectations Dosi, Anar (Elsevier BV, 2018-10-01) The present paper is devoted to classification of multinormed W*-algebras in terms of their bounded parts. We obtain a precise description of the bornological predual of a multinormed W*-algebra, which is reduced to a multinormed noncommutative L-1-space in the semifinite case. A multinormed noncommutative L-2-space is obtained as the union space of a commutative domain in a semifinite von Neumann algebra. Multinormed W*-algebras of type I are described as locally bounded decomposable (unbounded) operators ...
Integral criteria for oscillation of third order nonlinear differential equations AKTAŞ, MUSTAFA FAHRİ; Tiryaki, Aydın; Zafer, Ağacık (Elsevier BV, 2009-12-15) In this paper we are concerned with the oscillation of third order nonlinear differential equations of the form
Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping Tiryaki, A; Zafer, Ağacık (Elsevier BV, 2005-01-01) The paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping terms. Several new interval oscillation criteria are established for such a class of differential equations under quite general assumptions. Examples are also given to illustrate the results. In particular, it is shown that under some very mild conditions on k(1), k(2), and f the equation
Banach-stone theorem for Banach lattice valued continuous functions Ercan, Z.; Onal, S. (American Mathematical Society (AMS), 2007-01-01) Let Chi and Upsilon be compact Hausdor spaces, Epsilon be a Banach lattice and F be an AM space with unit. Let pi : C(X, E) -> C(Upsilon, F) be a Riesz isomorphism such that 0 not subset of f (X) if and only if 0 not subset of pi (f)(Upsilon) for each f is an element of C(Chi, Epsilon). We prove that Chi is homeomorphic to Upsilon and Epsilon is Riesz isomorphic to F. This generalizes some known results.
On the reduction principle for differential equations with piecewise constant argument of generalized type Akhmet, Marat (Elsevier BV, 2007-12-01) In this paper we introduce a new type of differential equations with piecewise constant argument (EPCAG), more general than EPCA [K.L. Cooke, J. Wiener, Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984) 265-297; J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993]. The Reduction Principle [V.A. Pliss, The reduction principle in the theory of the stability of motion, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1964) 1297...

Citation Formats

Z. Ercan and S. Onal, “Kakutani-Krein compact space of the CDw(X)-space in terms of X x {0,1},” JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, pp. 611–631, 2006, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/65671.