Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T20:25:05.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme Point Methods and Banach-Stone Theorems

Part of: Linear function spaces and their duals Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

Hasan Al-Halees  and
Richard J. Fleming
Hasan Al-Halees
Affiliation:
Department of Mathematics Saginaw Valley State UniversitySaginaw MI 48710USA e-mail: [email protected]
Richard J. Fleming
Affiliation:
Department of Mathematics Central Michigan UniversityMt. Pleasant MI 48859USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of C0( Q, X) into C0(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.

MSC classification

Secondary: 46B04: Isometric theory of Banach spaces 46E40: Spaces of vector- and operator-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 1 , August 2003 , pp. 125 - 143
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Al-Halees, H., Banach-Stone theorems for nice operators on Banach function modules (Ph.D. Thesis, Central Michigan University, 2001).Google Scholar
[2]Banach, S., Theorie des operations lineares (Chelsea, Warsaw, 1932).Google Scholar
[3]Behrends, E., M structure and the Banach-Stone theorem, Lecture Notes in Math. 736 (Springer, Berlin, 1979).Google Scholar
[4]Brosowski, B. and Deutsch, F., ‘On some geometric properties of suns’, J.Approx. Theory10(1974), 245–267.CrossRefGoogle Scholar
[5]Cambern, M., ‘On mapping of spaces of functions with values in a Banach space’, Duke Math. J. 48 (1975), 91–98.Google Scholar
[6]Cambern, M., ‘A Holsztynski theorem for spaces of vector valued functions’, Studia Math. 63 (1978), 213–217.CrossRefGoogle Scholar
[7]Cengiz, B., ‘On extremely regular function spaces’, Pac. J. Math. 49 (1973), 335–338.Google Scholar
[8]Dunford, N. and Schwartz, J., Linear operators, Part I (Interscience, New York, 1958).Google Scholar
[9]Fleming, R. and Jamison, J., ‘Isometries on Banach spaces: a survey’, in: Analysis, geometry, and groups, a Riemann legacy volume (eds. Srivastava, H. and Rassias, T.) (Hadronic Press, Palm Harbor, Florida, 1993) pp. 52–123.Google Scholar
[10]Font, J., ‘Linear isometries between certain subspaces of continuous vector-valued functions’, Illinois J. Math. 42 (1998), 389–397.CrossRefGoogle Scholar
[11]Holsztynski, W., ‘Continuous mappings induced by isometries of spaces of continuous functions’, Studia Math. 26 (1966), 133–136.CrossRefGoogle Scholar
[12]Jeang, J. and Wong, N., ‘On the Banach-Stone problem’, preprint, 2001.Google Scholar
[13]Jerison, M., ‘The space of bounded maps into a Banach space’, Ann. of Math. 52 (1950), 309–327.CrossRefGoogle Scholar
[14]Leibowitz, G., Lectures on complex function algebras (Scott, Foresman, and Company, Glenview, Illinois, 1970).Google Scholar
[15]McDonald, J.,‘Isometries of function algebras’, Illinois J. Math. 17 (1973), 579–583.CrossRefGoogle Scholar
[16]Morris, P. and Phelps, R., ‘Theorems of Krein-Milman type for certain convex sets of operators’, Trans. Amer. Math. Soc. 150 (1970), 183–200.Google Scholar
[17]Novinger, W., ‘Linear isometries of subspaces of continuous functions’, Studia Math. 53 (1975), 273–276.Google Scholar
[18]Phelps, R., Lectures on Choquet's theorem, Lecture Notes in Math. 1757, 2nd Edition (Springer, Berlin, 2001).Google Scholar
[19]Stone, M. H., ‘Applications of the theory of boolean rings in topology, Trans. Amer. Math. Soc. 41 (1937), 375–481.Google Scholar
[20]Werner, D., ‘Extreme points in spaces of operators and vector-valued measures’, Proc. Amer. Math. Soc. 89 (1983), 1–9.Google Scholar