## Banach spaces whose duals contain $l_{1}(\Gamma )$ with applications to the study of dual $L_{1}(\mu )$ spaces

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- by C. Stegall PDF
- Trans. Amer. Math. Soc.
**176**(1973), 463-477 Request permission

## 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

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**176**(1973), 463-477 - MSC: Primary 46B05; Secondary 46E30
- DOI: https://doi.org/10.1090/S0002-9947-1973-0315404-3
- MathSciNet review: 0315404