((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 \(L_{1}\) (\(=L_{1}[0,1]\)) 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: \(\bullet\) \(X\) contains almost isometric copies of \((\bigoplus_{n=1}^{\infty} \ell_{\infty}^{n})_{\ell_1}\), \(\bullet\) for all \(\varepsilon>0\), \(X^{*}\) contains a \((1+\varepsilon)\)-complemented, \((1+\varepsilon)\)-isomorphic copy of \(L_{1}\), \(\bullet\) 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: \(\bullet\) for all \(\varepsilon>0\), there exists a \((1+\varepsilon)\)-quotient map \(T\colon X\rightarrow C(\Delta)\) so that \(T^{*}[C(\Delta)^{*}]\) is \((1+\varepsilon)\)-complemented in \(X^{*}\), where \(\Delta\) is the Cantor set.