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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.

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