Bounded linear operators between C^*-algebras

Let $u:A\to B$ be a bounded linear operator between two $C^*$-algebras $A,B$. The following result was proved by the second author. Theorem 0.1. There is a numerical constant $K_1$ such that for all finite sequences $x_1,\ldots, x_n$ in $A$ we have \leqalignno{&\max\left\{\left\|\left(\sum u(x_i)^* u(x_i)\right)^{1/2}\right\|_B, \left\|\left(\sum u(x_i) u(x_i)^*\right)^{1/2}\right\|_B\right\}&(0.1)_1\cr \le &K_1\|u\| \max\left\{\left\|\left(\sum x^*_ix_i\right)^{1/2}\right\|_A, \left\|\left(\sum x_ix^*_i\right)^{1/2}\right\|_A\right\}.} A simpler proof was given in [H1]. More recently an other alternate proof appeared in [LPP]. In this paper we give a sequence of generalizations of this inequality.