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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On weakly countably determined Banach spaces


Author: Sophocles Mercourakis
Journal: Trans. Amer. Math. Soc. 300 (1987), 307-327
MSC: Primary 46B20; Secondary 47B38, 54H05
MathSciNet review: 871678
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Abstract: For a topological space $ X$, let $ {C_1}(X)$ denote the Banach space of all bounded functions $ f:X \to {\mathbf{R}}$ such that for every $ \varepsilon > 0$ the set $ \{ x \in X:\vert f(x)\vert \geqslant \varepsilon \} $ is closed and discrete in $ X$, endowed with the supremum norm. The main theorem is the following: Let $ L$ be a weakly countably determined subset of a Banach space; then there exist a subset $ \Sigma '$ of the Baire space $ \Sigma $, a compact space $ K$, and a bounded linear one-to-one operator $ T:C(L) \to {C_1}(\Sigma ' \times K)$ that is pointwise to pointwise continuous. In the case where $ L$ is weakly analytic, $ \Sigma '$ can be replaced by $ \Sigma $. This theorem is connected with the basic result of Amir-Lindenstrauss on WCG Banach spaces and has corresponding consequences such as: the representation of Gulko (resp. Talagrand) compact spaces as pointwise compact subsets of $ {C_1}(\Sigma ' \times K)$ (resp. $ {C_1}(\Sigma \times K)$) (a compact space $ \Omega $ is called Gulko or Talagrand compact if $ C(\Omega )$ is WCD or a weakly $ K$-analytic Banach space); the characterization of WCD (resp. weakly $ K$-analytic) Banach spaces $ E$, using one-to-one operators from $ {E^{\ast}}$ into $ {C_1}(\Sigma ' \times K)$ (resp. $ {C_1}(\Sigma \times K)$); and the existence of equivalent "good" norms on $ E$ and $ {E^{\ast}}$ simultaneously.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0871678-1
PII: S 0002-9947(1987)0871678-1
Article copyright: © Copyright 1987 American Mathematical Society