The factorisation property of l∞(Xk).

In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by (e(k,j)*)j, for k ∈ N, let Z = ℓ∞(Xk : k ∈ N) be their l∞-sum, and let T : Z → Z be a bounded linear operator with a large diagonal, i.e., infk,j|e*(k,j)(T(e(k,j))| > 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. [ABSTRACT FROM AUTHOR]