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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Banach-Stone property and the weak Banach-Stone property in three-dimensional spaces

Author: Michael Cambern
Journal: Proc. Amer. Math. Soc. 67 (1977), 55-61
MSC: Primary 46E40
MathSciNet review: 0461126
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Abstract: Let X and Y be compact Hausdorff spaces, E a Banach space, and $ C(X,E)$ the space of continuous functions on X to E. E has the weak Banach-Stone property if, whenever $ C(X,E)$ and $ C(Y,E)$ are isometric, then X and Y are homeomorphic. E has the Banach-Stone property if the descriptive as well as the topological conclusions of the Banach-Stone theorem for scalar functions remain valid in the case of isometries of $ C(X,E)$ onto $ C(Y,E)$. These two properties were first studied by M. Jerison, and it we later shown that every space E found by Jerison to have the weak Banach-Stone property actually has the Banach-Stone property, thus raising the question of whether the two properties are distinct. Here we characterize all three-dimensional spaces with the weak Banach-Stone property, and, in so doing, show the properties to be distinct.

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Article copyright: © Copyright 1977 American Mathematical Society

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