Spaces of operators, \(c_0\) and \(l^1\) (Q1768081)

A classical result of Pelczynski says that if the dual \(X^\ast\) of a Banach space \(X\) contains a copy of \(c_0\), then \(X^\ast\) already contains a (1-complemented) copy of \(\ell_\infty\) and \(X\) contains a complemented copy of \(\ell_1\). Thus, for Banach spaces \(X\) and \(Y\), \(L(X,Y^{\ast\ast})\) contains a copy of \(c_0\) iff it contains a copy of \(\ell_\infty\) iff \(X\otimes_\pi Y^\ast\) contains a complemented copy of \(\ell_1\). The authors observe that the embedding of \(\ell_1\) into \(X\otimes_\pi Y^\ast\) maps each unit vector in \(\ell_1\) to a finite tensor. The important result of this short note is, however, that we cannot substitute \(Y^{\ast\ast}\) by \(Y\) in the last equivalence of the above theorem. More precisely, whenever \(Y\) is such that \(Y^\ast\) is separable, \(\ell_1\) embeds complementably into \(Y^\ast\) but \(c_0\) does not embed as a subspace of \(Y\), then there exist a Banach space \(X\) and an isomorphic embedding \(J\) of \(\ell_1\) into \(X\otimes_\pi Y^\ast\) such that its image is a complemented subspace and each unit vector is taken to a finite rank tensor, but \(L(X,Y)\) does not contain a copy of \(c_0\). This completes the results from \textit{P. Lewis} [Stud. Math. 145, 213--218 (2001; Zbl 0986.46011)].