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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}$.

References [Enhancements On Off] (What's this?)

  • [1] Ehrhard Behrends, 𝑀-structure and the Banach-Stone theorem, Lecture Notes in Mathematics, vol. 736, Springer, Berlin, 1979. MR 547509
  • [2] Ehrhard Behrends, On the geometry of spaces of 𝐶₀𝐾-valued operators, Studia Math. 90 (1988), no. 2, 135–151. MR 954168
  • [3] Michael Cambern and Peter Greim, Mappings of continuous functions on hyper-Stonean spaces, Acta Univ. Carolin. Math. Phys. 28 (1987), no. 2, 31–40. 15th winter school in abstract analysis (Srní, 1987). MR 932737
  • [4] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
  • [5] Ryszard Engelking, General topology, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR 0500780
  • [6] Meyer Jerison, The space of bounded maps into a Banach space, Ann. of Math. (2) 52 (1950), 309–327. MR 0036942
  • [7] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR 0070144
  • [8] I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515–531. MR 0370466
  • [9] I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), no. 4, 735–750. MR 0390721

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DOI: https://doi.org/10.1090/S0002-9939-1989-0968623-4
Article copyright: © Copyright 1989 American Mathematical Society