$(1+)$-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 Pełczyński's classical work on dual Banach spaces containing L 1 (= L 1 [0, 1]) and the Hagler-Stegall characterisation of dual spaces containing complemented copies of L 1 . We prove 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 copies of (Moreover, if X is separable, one may add the following assertion:• for all ε > 0, there exists a (1 + ε)-quotient map T : X → C(∆) so that T * [C(∆) * ] is (1 + ε)-complemented in X * , where ∆ is the Cantor set.