Factorization through Matrix Spaces for Finite Rank Operators between C∗-algebras

0. Introduction. In this paper we consider factorizations of finite rank operators through finite-dimensional C∗-algebras. We are interested in factorization norms involving either the completely bounded norm ‖ ‖cb or Haagerup’s decomposable norm ‖ ‖dec (see [11]). Let us denote byMn the C∗-algebra of all n×n matrices with complex entries. Let A and B be two C∗-algebras, and let us consider a finite rank bounded operator u : A→ B. Then for n large enough, say n≥ rk(u), we may write factorizations of the form u= βα, for some bounded operators