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2016 | OriginalPaper | Chapter

1. Introduction

Authors : H. G. Dales, F. K. Dashiell Jr., A. T.-M. Lau, D. Strauss

Published in: Banach Spaces of Continuous Functions as Dual Spaces

Publisher: Springer International Publishing

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Abstract

In this chapter, we shall begin by introducing some basic notations. This will be followed by a discussion in \(\S\) 1.4 of some topological concepts, including those of locally compact spaces and Stonean spaces. We shall later frequently refer to the Stone–Čech compactification β X of a completely regular space X, and this is introduced in \(\S\) 1.5. In \(\S\) 1.6, we shall prove Gleason’s theorem characterizing projective topological spaces as the Stonean spaces, and, in \(\S\) 1.7, we shall also recall some basic theory of Boolean algebras, generalizing this slightly to cover Boolean rings; we shall discuss the Stone space of a Boolean ring and give various important examples of Boolean rings.

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next chapter Banach Spaces and Banach Lattices
Footnotes
1
An example of C. H. Dowker shows that the converse fails; see [99, 6.2.20] and [112, 16M(Δ 1)]. In fact there is a (non-normal) zero-dimensional, locally compact space X such that βX is not zero-dimensional [236]; this contradicts a sentence in the book [155, p. 169] of Kelley.
 
Literature
1.
Y. A. Abramovich, C. D. Aliprantis, in An Invitation to Operator Theory. Graduate Studies in Mathematics, vol. 50 (American Mathematical Society, Providence, 2002)
4.
E. M. Alfsen, Compact Convex Sets and Boundary Integtrals. Ergebnisse der Mathematik und ihrer Granzgebiete, Band 57 (Springer, New York, 1971)
5.
C. D. Aliprantis, O. Burkinshaw, Positive Operators, 2nd printing (with revised section numbers, page numbers, and references) (Springer, Dordrecht, 2006)
23.
W. G. Bade, The Space of all Continuous Functions on a Compact Hausdorff Space (University of California, Berkeley, 1957). Library call no. QA689.B16
24.
W. G. Bade, The Banach Space C(S). Lecture Note Series, vol. 26 (Matematisk Institut, Aarhus Universitët, Aarhus, 1971)
28.
R. Balbes, P. Dwinger, Distributive Lattices (University of Missouri Press, Columbia, 1974)MATH
29.
B. Balcar, P. Simon, Disjoint refinements, in Handbook of Boolean Algebras, vol. 2, ed. by J. D. Monk (North-Holland/Elsevier, Amsterdam, 1989), pp. 334–386
34.
J. F. Berglund, H. D. Junghenn, P. Milnes, Analysis on Semigroups; Function Spaces, Compactifications, Representations. Canadian Mathematical Society Series of Monographs and Advanced Texts (Wiley, New York, 1989)MATH
59.
62.
W. W. Comfort, S. Negrepontis, Continuous Pseudometrics. Lecture Notes in Pure and Applied Mathematics, vol. 14 (Marcel Dekker, New York, 1975)
68.
H. G. Dales, Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, vol. 24 (Clarendon Press, Oxford, 2000)
71.
H. G. Dales, A. T.-M. Lau, D. Strauss, Banach algebras on semigroups and on their compactifications. Mem. American Math. Soc. 205, 165 (2010)MathSciNetMATH
74.
H. G. Dales, W. H. Woodin, An Introduction to Independence for Analysts. London Mathematical Society Lecture Note Series, vol. 115 (Cambridge University Press, Cambridge, 1987)
84.
J. Diestel, J. H. Fourie, J. Swart, The Metric Theory of Tensor Products. Grothendieck’s Résumé Revisited (American Mathematical Society, Providence, 2008)
87.
91.
J. Dixmier, Sur certains espaces considérés par M. H. Stone. Summa Brasiliensis Math. 2, 151–182 (1951)MathSciNetMATH
94.
N. Dunford, J. T. Schwartz, Linear Operators, Part I: General Theory (Interscience Publishers, New York, 1957)MATH
96.
P. Dwinger, Introduction to Boolean Algebras, 2nd edn. (Physica-Verlag, Würtzburg, 1971)MATH
99.
R. Engelking, General Topology. Monografie Matematyczne, vol. 60 (Polish Scientific Publishers, Warsaw, 1977). Revised and completed edition (Heldermann Verlag, Berlin, 1989)
104.
V. P. Fonf, J. Lindenstrauss, R. R. Phelps, Infinite dimensional convexity, in Handbook of the Geometry of Banach Spaces, vol. 1, ed. by W. B. Johnson, J. Lindenstrauss (North Holland/Elsevier, Amsterdam, 2001), pp. 599–670CrossRef
111.
L. Gillman, M. Henriksen, Rings of continuous fumctions in which every finitely generated ideal is principal. Trans. American Math. Soc. 82, 366–391 (1956)MathSciNetCrossRefMATH
112.
L. Gillman, M. Jerison, Rings of Continuous Functions (van Nostrand Reinhold, New York, 1960). Reprinted as Graduate Texts in Mathematics, vol. 43 (Springer, New York, 1976)
113.
S. Givant, P. R. Halmos, Introduction to Boolean Algebras (Springer, New York, 2009)MATH
114.
117.
118.
K. R. Goodearl, Partially Ordered Abelian Groups with Interpolation. Mathematical Surveys and Monographs, vol. 20 (American Mathematical Society, Providence, 1986)
132.
138.
E. Hewitt, K. Stromberg, Real and Abstract Analysis (Springer, New York, 1975)MATH
139.
N. Hindman, D. Strauss, Algebra in the Stone–Čech Compactification, Theory and Applications (Walter de Gruyter, Berlin, 1998). Second revised and extended edition (2012)
141.
J. G. Hocking, G. S. Young, Topology (Addison-Wesley, Reading, 1961). Currently available in unabridged, corrected re-publication, Dover Books on Mathematics (Dover Publications, Mineola/New York, 1988)
142.
145.
T. Jech, Set Theory. The Third Millenium Edition, Revised and Expanded. Springer Monographs in Mathematics (Springer, Berlin, 2002). Corrected 4th printing (2006)
155.
J. L. Kelley, General Topology (Van Nostrand, Toronto, 1955)MATH
159.
S. Koppelberg, General theory of Boolean algebras, in Handbook of Boolean Algebras, vol. I, ed. by J. D. Monk, R. Bonnet (North Holland/Elsevier, Amsterdam, 1989)
164.
180.
W. A. J. Luxemburg, A. C. Zaanan, Riesz Spaces, vol. I (North Holland, Amsterdam, 1971)
184.
185.
J. van Mill, An introduction to β ω, in Handbook of Set Theoretic Topology, ed. by K. Kunen, J. E. Vaughan (North Holland/Elsevier, Amsterdam, 1984), pp. 503–567
191.
203.
R. S. Pierce, Countable Boolean algebras, in Handbook of Boolean Algebras, vol. III, ed. by J. D. Monk, R. Bonnet (North-Holland/Elsevier, Amsterdam, 1989), pp. 775–876
209.
C. A. Rogers, J. E. Jayne et al., Analytic Sets (Academic, London, 1980)MATH
210.
I. Rosenholtz, Another proof that any compact metric space is the continuous image of the Cantor set. American Math. Monthly 83, 646–647 (1976)MathSciNetCrossRefMATH
212.
H. P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37, 13–36 (1970)MathSciNetMATH
215.
H. P. Rosenthal, The Banach spaces C(K), in Handbook of the Geometry of Banach Spaces, vol. 2, ed. by W. B. Johnson, J. Lindenstrauss (North Holland/ Elsevier, Amsterdam, 2003), pp. 1547–1602CrossRef
217.
W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill, New York, 1986)MATH
223.
225.
Z. Semadeni, Banach Spaces of Continuous Functions. Monografie Matematyczne, vol. 55 (Instytut Matematyczny Polskiej Akademii Nauk, Warsaw, 1971)
228.
S. M. Srivastava, A Course on Borel Sets. Graduate Texts in Mathematics, vol. 180 (Springer, New York, 1998)
231.
M. H. Stone, The theory of representations of Boolean algebras. Trans. American Math. Soc. 40, 37–111 (1936)MathSciNetMATH
232.
236.
J. Terasawa, Spaces N ∪ R need not be strongly 0-dimensional. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astonom. Phys. 25, 279–281 (1977)MathSciNetMATH
237.
S. Todorcevic, Topics in Topology. Lecture Notes in Mathematics, vol. 1652 (Springer, Berlin, 1997)
239.
241.
245.
Metadata
Title
Introduction
Authors
H. G. Dales
F. K. Dashiell Jr.
A. T.-M. Lau
D. Strauss
Copyright Year
2016
Book
Banach Spaces of Continuous Functions as Dual Spaces

Print ISBN: 978-3-319-32347-3

Electronic ISBN: 978-3-319-32349-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-32349-7_1

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