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 : E → F, \({y \in Y}\) , 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.
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Research of D. H. Leung was partially supported by AcRF project no. R-146-000-086-112.
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Leung, D.H., Tang, WK. Banach–Stone Theorems for maps preserving common zeros. Positivity 14, 17–42 (2010). https://doi.org/10.1007/s11117-008-0002-3
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DOI: https://doi.org/10.1007/s11117-008-0002-3