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On uniformly differentiable mappings from $\ell_\infty(Γ)$

In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $Γ$ is an infinite index set, and $T:\ell_\infty(Γ)\to X$ is a bounded linear operator such that $\inf_{γ\inΓ}\|T(e_γ)\|>0$ then $T$ acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ of the same cardinality as $Γ$. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of $Γ$, we show that if ${F}:B_{\ell_\infty(Γ)}\to X$ is uniformly differentiable and such that $\inf_{γ\inΓ}\|{F}(e_γ){-F(0)}\|>0$ then there exists $x\in B_{\ell_\infty(Γ)}$ such that $d{F}(x)[\cdot]$ is a bounded linear operator which acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ of the same cardinality as $Γ$.