Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T11:09:08.430Z Has data issue: false hasContentIssue false
  • English
  • Français

Diameter-preserving linear bijections of function spaces

Part of: Special classes of linear operators Linear function spaces and their duals Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  09 April 2009

T. S. S. R. K. Rao  and
A. K. Roy
T. S. S. R. K. Rao
Affiliation:
Statistics and Mathematics Unit Indian Statistical Institute R. V. College PostBangalore 560 059India e-mail: [email protected]
A. K. Roy
Affiliation:
Statistics and Mathematics Unit Indian Statistical Institute203 B. T. Road Calcutta 700 035India 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.

In this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

MSC classification

Secondary: 46J10: Banach algebras of continuous functions, function algebras 46E40: Spaces of vector- and operator-valued functions 47B38: Operators on function spaces (general)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 3 , June 2001 , pp. 323 - 336
Copyright
Copyright © Australian Mathematical Society 2001

References

[AE]Asimow, L. and Ellis, A. J., Convexity theory and its applications in functional analysis, London Math. Soc. Monographs 16 (Academic Press, London, 1980).Google Scholar
[B]Behrends, E., M-structures and the Banach-Stone theorem, Lecture Notes in Math. 736 (Springer, Berlin, 1979).Google Scholar
[C]Sanchez, F. Cabello, ‘Diameter preserving linear maps and isometries’, Arch. Math. 73 (1999), 373–379.Google Scholar
[D]Diestel, J., Sequences and series in Banach spaces, Graduate Texts in Math. 92 (Springer, Berlin, 1984).Google Scholar
[DU]Diestel, J. and Uhl, J. J., Vector measures, Amer. Math. Soc. Surveys 15 (Providence, RI, 1977).Google Scholar
[ES]Ellis, A. J. and So, W. S., ‘Isometries and complex state spaces of uniform algebras’, Math. Z. 195 (1987), 119–125.Google Scholar
[FJ]Fleming, R. J. and Jamison, J. E., ‘Isometries on Banach spaces: A survey’, in: Analysis, geometry and groups: A Riemann legacy volume (Hadronic Press, Palm Harbor, 1993) pp. 52–123.Google Scholar
[GJ]Gillman, L. and Jerison, M., Rings of continuous functions, Graduate Texts in Math. 43 (Springer, Berlin, 1976).Google Scholar
[GU]González, F. and Uspenskij, V. V., ‘On homeomorphisms of groups of integer-valued functions’, Extracta Math. 14 (1999), 19–29.Google Scholar
[GM]Györy, M. and Molnar, L., ‘Diameter preserving linear bijections of C(X)’, Arch. Math. 71 (1998), 301–310.Google Scholar
[H]Holmes, R. B., Geometric functional analysis and its applications, Graduate Texts in Math. 24 (Springer, Berlin, 1975).CrossRefGoogle Scholar
[L]Lazar, A. J., ‘Affine products of simplexes’, Math. Scand. 22 (1968), 165–175.Google Scholar
[Li]Lima, Å., Oja, E., Rao, T. S. S. R. K. and Werner, D., ‘Geometry of operator spaces’, Michigan Math. J. 41 (1994), 473–490.Google Scholar
[R1]Rao, T. S. S. R. K., ‘Isometries of AC(K)’, Proc. Amer. Math. Soc. 85 (1982), 544–546.Google Scholar
[R2]Rao, T. S. S. R. K., ‘M-structure and the space A (K, E)’, Rend. Mat. Appl. 15 (1995), 153–160.Google Scholar
[R3]Rao, T. S. S. R. K., ‘Denting and strongly extreme points in the unit ball of spaces of operators’, Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999), 75–85.Google Scholar
[Ru]Ruess, W., ‘Duality and geometry of spaces of compact operators’, in: Surveys and recent results III. Proc. Pederborn Conference on Functional Analysis (eds. Bierstedt, K. and Fuchssteiner, B.). North-Holland Math. Studies 90 (North-Holland, Amsterdam, 1984)pp. 59–78.Google Scholar
[RW]Ruess, W. and Werner, D., ‘Structural properties of operator spaces’, Acta Univ. Carol. Math. Phys. 28 (1987), 127–136.Google Scholar
[W]Walker, R. C., The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete Band 83 (Springer, Berlin, 1974).Google Scholar