The factorization property of \(\ell^\infty(X_k)\)

In this paper we consider the following problem: Let \(X_k\), be a Banach space with a normalized basis \((e_{(k,j)})_j\), whose biorthogonals are denoted by \((e_{(k,j)}^*)_j\), for \(k\in\mathbb{N}\), let \(Z=\ell^\infty(X_k:k\in\mathbb{N})\) be their \(\ell^\infty\)-sum, and let \(T:Z\to Z\) be a bounded linear operator, with a large diagonal, i.e. $$\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.$$ Under which condition does the identity on \(Z\) factor through \(T\)? The purpose of this paper is to formulate general conditions for which the answer is positive.