Pisier–Ringrose inequality

In mathematics, Pisier–Ringrose inequality is an inequality in the theory of C*-algebras which was proved by Gilles Pisier in 1978 affirming a conjecture of John Ringrose. It is an extension of the Grothendieck inequality.

Statement
Theorem. If $$\gamma$$ is a bounded, linear mapping of one C*-algebra $$\mathfrak{A}$$ into another C*-algebra $$\mathfrak{B}$$, then
 * $$\left\|\sum_{j=1}^n \gamma(A_j)^* \gamma(A_j) + \gamma(A_j) \gamma(A_j)^*\right\| \le 4 \|\gamma \|^2 \left\| \sum_{j=1}^n A_j^*A_j + A_j A_j^* \right\|$$

for each finite set $$\{ A_1, A_2, \ldots, A_n \}$$ of elements $$A_j$$ of $$\mathfrak{A}$$.