# Banach–Stone Theorems for maps preserving common zeros

Banach–Stone Theorems for maps preserving common zeros 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: $$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z})$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Banach–Stone Theorems for maps preserving common zeros

, Volume 14 (1) – Jan 21, 2009
26 pages

/lp/springer_journal/banach-stone-theorems-for-maps-preserving-common-zeros-CwRFTEVaay
Publisher
Springer Journals
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-0002-3
Publisher site
See Article on Publisher Site

### 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: $$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z})$$ 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.

### Journal

PositivitySpringer Journals

Published: Jan 21, 2009

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