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

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