Weak compactness in projective tensor products of Banach spaces (Q535124)

The author proves that the main result in [\textit{J.~Diestel} and \textit{D.~Puglisi}, J. Math.\ Anal.\ Appl.\ 350, No.~2, 508--513 (2009; Zbl 1170.46066)] is false. In that paper, the authors attempted to give a description of the weakly compact sets in the projective tensor product of two Banach spaces. It was pointed out in the above review that their proof is not conclusive, and the present article shows that the gap cannot be fixed. Indeed, a consequence of the Diestel-Puglisi condition is that a weakly compact subset of \(X \widehat\otimes_\pi Y\) is contained in the closed convex hull of some set \(A \otimes B\), with \(A\subset X\) and \(B\subset Y\) weakly compact. The latter is obviously true if \(X\) and \(Y\) are reflexive or if \(X\widehat\otimes_\pi Y\) has the Schur property. The author proves that it fails, though, for many classical Banach spaces, e.g., for \(X=L_1[0,1]\) and \(Y=c_0\), and he suggests the problem to find a nontrivial example where it holds.