Grothendieck property of non-reflexive injective tensor products

Determine whether the injective tensor product of two Banach spaces is necessarily non-Grothendieck whenever it fails to be reflexive, thereby resolving the stated conjecture about non-reflexive injective tensor products.

Background

The introduction discusses indecomposability phenomena for tensor products and notes that the projective tensor product of two Banach spaces can be Grothendieck only when at least one factor is reflexive. It then presents the corresponding unresolved conjecture for the injective tensor product: failure of reflexivity should imply failure of the Grothendieck property. The paper proves related results for Wittstock tensor products of Banach lattices under positivity and approximation-property assumptions, but does not resolve the general Banach-space conjecture.

References

For general Banach spaces, it is known that the projective tensor product $E\widetilde{\otimes}{\pi}F$ is Grothendieck only if $E$ or $F$ is reflexive (see Proposition 5.3.1), and conjectured that the injective tensor product $E\widetilde{\otimes}{\varepsilon}F$ is not Grothendieck if it fails to be reflexive (see p. 1158).

— Complemented Copies of $c_{0}$ in Positive Tensor Products of Banach Lattices  (2608.24834 - Melnikov, 25 Aug 2026) in Section 1, Introduction