On banach spaces which containl
1(τ) and types of measures on compact spaces |
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Authors: | Richard Haydon |
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Institution: | (1) Brasenose College, Oxford, England |
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Abstract: | Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic
properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL
1. Let τ be a regular cardinal satisfying the hypothesis that κω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto 0, 1]τ. 2) A Banach spaceX has a subspace isomorphic tol
1(τ) if and only ifX
∗ has a subspace isomorphic toL
1({0, 1}τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl
1 does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is
a Banach spaceX which contains no copy ofl
1 (ω1), while the unit ball ofX
∗ is not weakly∗ sequentially compact. |
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Keywords: | |
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