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A lattice-valued Banach-Stone theorem. (English) Zbl 1027.46025
Let \(X_1\) and \(X_2\) be compact Hausdorff spaces and let \(C(X_i)\) (\(i=1,2\)) denote the space of real-valued continuous functions on \(X_i\) equipped with supremum norm. The deduction of topological affinities between the spaces \(X_1\) and \(X_2\) from certain algebraic or geometric relations between \(C(X_1)\) and \(C(X_2)\) has been widely treated in the literature, being the Banach-Stone theorem the first and most inspiring result. The authors deal with this type of questions for spaces of continuous functions that take values in a Banach lattice. Among others, their main result is the following: let \(X_1\), \(X_2\) be compact Hausdorff spaces and let \(E\) be a Banach lattice. Suppose there is a Riesz isomorphism \(\Phi : C(X_1,E)\longrightarrow C(X_2,\mathbb R)\) such that \(\phi(f)\) has no zeros if \(f\) has none. Then \(X_1\) and \(X_2\) are homeomorphic and \(E\) and \(\mathbb R\) are Riesz isomorphic.

46E05 Lattices of continuous, differentiable or analytic functions
46B42 Banach lattices
54C35 Function spaces in general topology
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