A renorming characterization of Banach spaces containing $\ell_1(\kappa)$

TL;DR

This paper extends Godefroy's renorming characterization of Banach spaces containing  to larger uncountable cardinals, providing a new criterion for the existence of () subspaces via equivalent norms.

Contribution

It generalizes Godefroy's result to uncountable cardinals, answering an open question about characterizations of () subspaces in larger Banach spaces.

Findings

Characterization of () in Banach spaces for uncountable 

Existence of equivalent norms with specific properties for () subspaces

Counterexample showing Godefroy's result cannot be improved for countable case

Abstract

A result of G. Godefroy asserts that a Banach space $X$ contains an isomorphic copy of $\ell_1$ if and only if there is an equivalent norm $|||\cdot|||$ such that, for every finite-dimensional subspace $Y$ of $X$ and every $\varepsilon>0$, there exists $x\in S_X$ so that $|||y+r x|||\geq (1-\varepsilon)(|||y|||+\vert r\vert)$ for every $y\in Y$ and every $r\in\mathbb R$. In this paper we generalize this result to larger cardinals, showing that if $\kappa$ is an uncountable cardinal then a Banach space $X$ contains a copy of $\ell_1(\kappa)$ if and only if there is an equivalent norm $|||\cdot|||$ on $X$ such that for every subspace $Y$ of $X$ with $dens(Y)<\kappa$ there exists a norm-one vector $x$ so that $||| y+r x|||=|||y|||+\vert r\vert$ whenever $y\in Y$ and $r\in\mathbb{R}$. This result answers a question posed by S. Ciaci, J. Langemets and A. Lissitsin, where the authors wonder whether the previous statement holds for infinite succesor cardinals. We also show that, in the countable case, the result of Godefroy cannot be improved to take $\varepsilon=0$.