Advertisement

Positivity

, Volume 14, Issue 1, pp 17–42 | Cite as

Banach–Stone Theorems for maps preserving common zeros

  • Denny H. Leung
  • Wee-Kee Tang
Article

Abstract

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) =  S y (f (h(y))) for a family of linear operators S y : EF, \({y \in Y}\) , and a function h: YX. In this paper, we consider maps having the property:
$$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $$
where Z(f) =  {f =  0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Önal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C *-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that
$$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $$
Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C *-isomorphic) to F. Some results concerning the continuity of T are also obtained.

Mathematics Subject Classification (2000)

Primary 47B38 Secondary 46E40 46H40 47B33 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Araujo J.: Realcompactness and spaces of vector-valued continuous functions. Fundam. Math. 172, 27–40 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Araujo J.: Realcompactness and Banach–Stone theorems. Bull. Belg. Math. Soc. Simon Stevin 11, 247–258 (2004)MATHMathSciNetGoogle Scholar
  3. 3.
    Araujo J.: Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity. Adv. Math. 187, 488–520 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Araujo J.: The noncompact Banach–Stone theorem. J. Operator Theory 55, 285–294 (2006)MATHMathSciNetGoogle Scholar
  5. 5.
    Araujo J., Beckenstein E., Narici L.: Biseparating maps and homoemorphic real-compactifications. J. Math. Anal. Appl. 192, 258–265 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Banach, S.: Théorie des Opérations Lineaires, Warszowa 1932. Reprinted, Chelsea Publishing Company, New York (1963)Google Scholar
  7. 7.
    Behrends E.: M-structure and the Banach–Stone Theorem. Springer, Berlin (1978)Google Scholar
  8. 8.
    Chen J.-X., Chen Z.-L., Wong N.-C.: A Banach–Stone Theorem for Riesz isomorphisms of Banach lattices. Proc. Am. Math. Soc. 136, 3869–3874 (2008)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dugundji J.: Topology Allyn and Bacon, Inc.. Allyn and Bacon, Inc., Boston (1966)MATHGoogle Scholar
  10. 10.
    Ercan Z., Önal S.: Banach–Stone Theorem for Banach lattice valued continuous functions. Proc. Am. Math. Soc. 135, 2827–2829 (2007)MATHCrossRefGoogle Scholar
  11. 11.
    Ercan Z., Önal S.: The Banach–Stone Theorem revisited. Topol. Appl. 155, 1800–1803 (2008)MATHCrossRefGoogle Scholar
  12. 12.
    Garrido M.I., Jaramillo J.A.: Variations on the Banach–Stone theorem. Extracta Math. 17, 351–383 (2002)MATHMathSciNetGoogle Scholar
  13. 13.
    Gau H.-W., Jeang J.-S., Wong N.-C.: Biseparating linear maps between continuous vector-valued function spaces. J. Aust. Math. Soc. 74, 101–109 (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gillman, L., Jerison, M.: Rings of continuous functions. Graduate Texts in Mathematics, No. 43. Springer, New York (1976)Google Scholar
  15. 15.
    Gelfand I., Kolmogorov A.: On rings of continuous functions on topological spaces. Dokl. Akad. Nauk. SSSR 22, 11–15 (1939)Google Scholar
  16. 16.
    Jeang J.-S., Wong N.-C.: On the Banach–Stone problem. Studia Math. 155, 95–105 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kaplansky I.: Lattices of continuous functions. Bull. Am. Math. Soc. 53, 617–623 (1947)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lang, S.: Real Analysis, 2nd edn. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA (1983)Google Scholar
  19. 19.
    Lin, Y.-F., Wong, N.-C.: The structure of compact disjointness preserving operators on continuous functions. Math. Nach. (to appear)Google Scholar
  20. 20.
    Schaefer H.H.: Topological Vector Spaces. The Macmillan Co., New York (1966)MATHGoogle Scholar
  21. 21.
    Stone M.H.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel / Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Mathematics and Mathematics Education, National Institute of EducationNanyang Technological UniversitySingaporeSingapore

Personalised recommendations