Banach-Stone Theorems for maps preserving common zeros

2009/06/01 by Denny H. Leung, Leung, Denny H., Wee-Kee Tang +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Advanced Topology and Set Theory #math.FA #msc:47B38

paper · pdf · doi:10.48550/arxiv.0906.0219

arxiv created 2009/06/01 · arxiv updated 2009/12/01

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 Tf(y) = S_y(f(h(y)) for a family of linear operators S_y : E → F, y ∈ Y, and a function h: Y → X. In this paper, we consider maps having the property: ∩^k_i=1Z(f_i) ≠∅\iff∩^k_i=1Z(Tf_i) ≠ ∅, 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) ≠∅\iff Z(Tf) ≠∅. 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.

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