On uniformly differentiable mappings from ℓ∞(Γ)

In 1970 Haskell Rosenthal proved that if X   is a Banach space, Γ is an infinite index set, and T:ℓ∞(Γ)→XT:ℓ∞(Γ)→X is a bounded linear operator such that infγ∈Γ⁡‖T(eγ)‖>0infγ∈Γ⁡‖T(eγ)‖>0 then T   acts as an isomorphism on ℓ∞(Γ′)ℓ∞(Γ′), for some Γ′⊂ΓΓ′⊂Γ 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:Bℓ∞(Γ)→XF:Bℓ∞(Γ)→X is uniformly differentiable and such that infγ∈Γ⁡‖F(eγ)−F(0)‖>0infγ∈Γ⁡‖F(eγ)−F(0)‖>0 then there exists x∈Bℓ∞(Γ)x∈Bℓ∞(Γ) such that dF(x)[⋅]dF(x)[⋅] is a bounded linear operator which acts as an isomorphism on ℓ∞(Γ′)ℓ∞(Γ′), for some Γ′⊂ΓΓ′⊂Γ of the same cardinality as Γ.