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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 Φ:C(X 1 ,E)C(X 2 ,) such that φ(f) has no zeros if f has none. Then X 1 and X 2 are homeomorphic and E and are Riesz isomorphic.

MSC:
46E05Lattices of continuous, differentiable or analytic functions
46B42Banach lattices
54C35Function spaces (general topology)