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

TL;DR

This paper provides a quantitative characterization of dual Banach spaces containing almost isometric and complemented copies of $L_1$, extending classical results with precise approximation and isomorphism conditions.

Contribution

It introduces a quantitative version of the Hagler--Stegall theorem, linking almost isometric copies of certain spaces in Banach spaces to complemented copies of $L_1$ in duals.

Findings

Equivalence between containing almost isometric copies of $(igoplus_{n=1}^{ abla} \, ext{ell}_ abla^n)_{ ext{ell}_1}$ and having complemented $L_1$ in the dual.

For all $ ext{ extvarepsilon}>0$, dual spaces contain $(1+ ext{ extvarepsilon})$-complemented, $(1+ ext{ extvarepsilon})$-isomorphic copies of $L_1$ and $C[0,1]^*$.

Separable spaces admit $(1+ ext{ extvarepsilon})$-quotient maps onto $C( ext{ extDelta})$ with complemented dual images.

Abstract

The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on Pe{\l}czy\'nski'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.