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

 

Banach spaces whose duals contain $ l\sb{1}(\Gamma )$ with applications to the study of dual $ L\sb{1}(\mu )$ spaces


Author: C. Stegall
Journal: Trans. Amer. Math. Soc. 176 (1973), 463-477
MSC: Primary 46B05; Secondary 46E30
MathSciNet review: 0315404
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Abstract: THEOREM I. If E is a separable Banach space such that $ E'$ has a complemented subspace isomorphic to $ {l_1}({\mathbf{\Gamma }})$ with $ {\mathbf{\Gamma }}$ uncountable then $ E'$ contains a complemented, $ \sigma (E',E)$ closed subspace isomorphic to $ M({\mathbf{\Delta }})$, the Radon measures on the Cantor set.

THEOREM II. If E is a separable Banach space such that $ E'$ has a subspace isomorphic to $ {l_1}({\mathbf{\Gamma }})$ with $ {\mathbf{\Gamma }}$ uncountable, then E contains a subspace isomorphic to $ {l_1}$,

THEOREM III. Let E be a Banach space. The following are equivalent:

(i) $ E'$ is isomorphic to $ {l_1}({\mathbf{\Gamma }})$;

(ii) every absolutely summing operator on E is nuclear;

(iii) every compact, absolutely summing operator on E is nuclear;

(iv) if X is a separable subspace of E, then there exists a subspace Y such that $ X \subseteq Y \subseteq E$ and $ Y'$ is isomorphic to $ {l_1}$.

THEOREM IV. If E is a $ {\mathcal{L}_\infty }$ space then (i) $ E'$ is isomorphic to $ {l_1}({\mathbf{\Gamma }})$ for some set $ {\mathbf{\Gamma }}$ or (ii) $ E'$ contains a complemented subspace isomorphic to $ M({\mathbf{\Delta }})$.

COROLLARY. If E is a separable $ {\mathcal{L}_\infty }$ space, then $ E'$ is (i) finite dimensional, or (ii) isomorphic to $ {l_1}$, or (iii) isomorphic to $ M({\mathbf{\Delta }})$.

COROLLARY. If $ {L_1}(\mu )$ is isomorphic to the conjugate of a separable Banach space, then $ {L_1}(\mu )$ is isomorphic to $ {l_1}$ or $ M({\mathbf{\Delta }})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0315404-3
PII: S 0002-9947(1973)0315404-3
Keywords: Dual $ {L_1}(\mu )$ spaces, $ {\mathcal{L}_\infty }$ spaces
Article copyright: © Copyright 1973 American Mathematical Society