On uniformly differentiable mappings from ${\ell }_{\infty }(\Gamma )$

Abstract

In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $\Gamma$ is an infinite index set, and $T:{\ell }_{\infty }(\Gamma )\to X$ is a bounded linear operator such that ${\inf }_{\gamma \in \Gamma }\Vert T(e_{\gamma })\Vert >0$ then $T$ acts as an isomorphism on ${\ell }_{\infty }({\Gamma }^{\prime })$, for some ${\Gamma }^{\prime }\subset \Gamma$ of the same cardinality as $\Gamma$. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of $\Gamma$, we show that if $F:B_{{\ell }_{\infty }(\Gamma )}\to X$ is uniformly differentiable and such that ${\inf }_{\gamma \in \Gamma }\Vert F(e_{\gamma })-F(0)\Vert >0$ then there exists $x\in B_{{\ell }_{\infty }(\Gamma )}$ such that $dF(x)[\cdot ]$ is a bounded linear operator which acts as an isomorphism on ${\ell }_{\infty }({\Gamma }^{\prime })$, for some ${\Gamma }^{\prime }\subset \Gamma$ of the same cardinality as $\Gamma$.