Banach-Stone Theorems for maps preserving common zeros

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) = Sy(f (h(y))) for a family of linear operators Sy: E → F, y ε Y, and a function h: Y → X. In this paper, we consider maps having the property:, 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, 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. © 2009 Birkhäuser Verlag Basel/Switzerland.