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

Banach–Stone Theorems for maps preserving common zeros

  • Denny H. Leung
  • Wee-Kee Tang


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 


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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

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