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

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 $\varepsilon=0$.