Banach-stone Theorems for Maps Preserving Common Zeros
نویسنده
چکیده
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 ∈ Y , and a function h : Y → X. In this paper, we consider maps having the property: (Z) ∩ki=1 Z(fi) 6= ∅ ⇐⇒ ∩ k i=1Z(Tfi) 6= ∅, 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) 6= ∅ ⇐⇒ Z(Tf) 6= ∅. 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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