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

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 $M({\mathbf {\Delta }})$.