Uniformly convex stalks and Kucher's ${\mathrm{\ell }}_{\mathrm{\infty }}(E)$ problem

Kucher asked whether ${\mathrm{\ell }}_{\mathrm{\infty }}(E)$ is a Grothendieck space for every super-reflexive Banach space $E$. We answer this question affirmatively by proving the stronger result that ${\mathrm{\ell }}_{\mathrm{\infty }}(E)$ has Pełczyński’s property $(V)$. Combined with the natural dual representation of ${\mathrm{\ell }}_{\mathrm{\infty }}(E)$, this yields that every non-weakly compact operator on ${\mathrm{\ell }}_{\mathrm{\infty }}(E)$ fixes a copy of ${\mathrm{\ell }}_{\mathrm{\infty }}$. The main ingredient is a general property-$(V)$ theorem for function modules over zero-dimensional compact Hausdorff spaces with stalks admitting a common modulus of uniform convexity. The proof is based on Gierz’s integral representation of functionals and the weak compactness of the associated canonical measures. In the case of ${\mathrm{\ell }}_{\mathrm{\infty }}(E)$, the relevant stalks are ultrapowers of $E$, and super-reflexivity provides the uniform geometric control required by the theorem. We also derive several consequences for operators on ${\mathrm{\ell }}_{\mathrm{\infty }}(E)$, including weak compactness into separable spaces, reflexivity of separable quotients, and compactness of operators into Schur spaces.