(1+)-complemented, (1+)-isomorphic copies of L_{1} in dual Banach spaces

Abstract

The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on the Hagler–Stegall characterisation of dual spaces containing complemented copies of L_{1}. As a corollary, we obtain the following quantitative version of the Hagler–Stegall theorem asserting that for a Banach space X, the following statements are equivalent:X contains almost isometric contains almost isometric copies of (bigoplus _{n=1}^{infty } ell _{infty }^{n})_{ell _1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of L_{1};for all varepsilon >0, X^{*} contains a (1+varepsilon )-complemented, (1+varepsilon )-isomorphic copy of C[0,1]^{*}. Moreover, if X is separable, one may add the following assertion:for all varepsilon >0, there exists a (1+varepsilon )-quotient map T:Xrightarrow C(Delta ) so that T^{*}[C(Delta )^{*}] is (1+varepsilon )-complemented in X^{*}, where Delta is the Cantor set