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

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 L1 (= L1[0, 1]) and the Hagler–Stegall characterisation of dual spaces containing complemented copies of L1. 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 ( ⊕ ∞ n=1 l n ∞ )l1 , • for all ε > 0, X∗ contains a (1 + ε)-complemented, (1 + ε)-isomorphic copy of L1, • for all ε > 0, X∗ contains a (1 + ε)-complemented, (1 + ε)-isomorphic copy of C[0, 1]∗. 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.