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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Facial characterizations of complex Lindenstrauss spaces

Authors: A. J. Ellis, T. S. S. R. K. Rao, A. K. Roy and U. Uttersrud
Journal: Trans. Amer. Math. Soc. 268 (1981), 173-186
MSC: Primary 46B10; Secondary 46A55
MathSciNet review: 628453
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Abstract: We characterize complex Banach spaces $A$ whose Banach dual spaces are ${L^1}(\mu )$ spaces in terms of $L$-ideals generated by certain extremal subsets of the closed unit ball $K$ of ${A^{\ast }}$. Our treatment covers the case of spaces $A$ containing constant functions and also spaces not containing constants. Separable spaces are characterized in terms of ${w^{\ast }}$-compact sets of extreme points of $K$, whereas the nonseparable spaces necessitate usage of the ${w^{\ast }}$-closed faces of $K$. Our results represent natural extensions of known characterizations of Choquet simplexes. We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of ${C_{\mathbf {C}}}(X)$ which are complex Lindenstrauss spaces.

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Article copyright: © Copyright 1981 American Mathematical Society