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Transactions of the American Mathematical Society
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0628453-7
PII: S 0002-9947(1981)0628453-7
Article copyright: © Copyright 1981 American Mathematical Society