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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Extreme points and $ l\sb{1}(\Gamma )$-spaces


Author: Nina M. Roy
Journal: Proc. Amer. Math. Soc. 86 (1982), 216-218
MSC: Primary 46B25; Secondary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1982-0667277-8
MathSciNet review: 667277
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Abstract: Let $ X$ be a nontrivial real Banach space and let $ {E_X}$ denote the set of extreme points of the closed unit ball $ B(X)$.

Theorem 1. $ X$ is an $ {l_1}(\Gamma )$-space if and only if (i) $ {\text{span}}(e)$ is an $ L$-summand in $ X$ for every $ e$ in $ {E_X}$ and (ii) $ B(X)$ is the norm closed convex hull of $ {E_X}$.

Theorem 2. Let $ X = {Y^ * }$. If (i) $ {\text{span}}(e)$ is an $ L$-summand in $ X$ for every $ e$ in $ {E_X}$ and (ii) $ \left\{ {e \in {E_X}:e(y) = 1} \right\}$ is countable for each $ y$ in $ Y$ with $ \left\Vert y \right\Vert = 1$, then $ X$ is an $ {l_1}(\Gamma )$-space.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0667277-8
Article copyright: © Copyright 1982 American Mathematical Society