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Proceedings of the American Mathematical Society

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



Isometries of spaces of weak $ *$ continuous functions

Authors: Michael Cambern and Krzysztof Jarosz
Journal: Proc. Amer. Math. Soc. 106 (1989), 707-712
MSC: Primary 46E40; Secondary 46B20
MathSciNet review: 968623
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Abstract: If $ X$ is a compact Hausdorff space and $ {E^*}$ a Banach dual, we denote by $ C(X,({E^*},{\sigma ^*}))$ the Banach space of continuous functions from $ X$ to $ {E^*}$ when the latter space is given its weak * topology. It is shown that if $ E_1^*,E_2^*$ have trivial centralizers and satisfy a topological condition introduced by Namioka and Phelps, then, given an isometry $ T$ mapping $ C(X_1^*,({E_1},{\sigma ^*}))$ onto $ C({X_2},(E_2^*,{\sigma ^*}))$ and given $ F \in C({X_1},(E_1^*,{\sigma ^*}))$, there exists a dense $ {G_\delta }$ in $ {X_2}$ on which

$\displaystyle (*)\quad (TF)(x) = U(x)F \circ \Phi (x),$

where $ \Phi $ is a homeomorphism of $ {X_2}$ onto $ {X_1}$ and $ U( \cdot )$ an operator-valued function independent of $ F$. If the $ E_i^*$ are separable then the $ U(x)$ are surjective isometries and $ x \to U(x)$ is continuous. When the $ {X_i}$ are metric the representation (*) holds on all of $ {X_2}$.

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Article copyright: © Copyright 1989 American Mathematical Society

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