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Mappings of continuous functions on hyperstonean spaces. (English) Zbl 0653.46036
If X is a compact Hausdorff space and \(E^*\) a Banach dual, denote by \(C(X,(E^*,\sigma^*))\) the Banach space of continuous functions on X to \(E^*\) when the latter space is given its weak* topology. Here we establish Banach-Stone theorems for spaces of this type, paralleling those for spaces of norm-continuous functions. If \(X_ 1,X_ 2\) are hyperstonean and \(E^*_ 1,E^*_ 2\) have one-dimensional centralizers then the existence of an isometry T of \(C(X_ 1,(E^*_ 1,\sigma^*))\) onto \(C(X_ 2,(E^*_ 2,\sigma^*))\) implies that \(X_ 1\) and \(X_ 2\) are homeomorphic. When in addition, the \(E^*_ i\) are separable, then they are isometrically isomorphic. In this case we also obtain an explicit description of T. Our description is (necessarily) more complicated than the one obtainable in the norm- continuous case, but we also give a necessary and sufficient condition on the \(E^*_ i\) which permits the description to be simplified.
46E40 Spaces of vector- and operator-valued functions
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