  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.
