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.^[1]^[2] If $γ$ is a bounded, linear mapping of one C*-algebra $𝔄$ into another C*-algebra $𝔅$ , then

‖ \sum_{j = 1}^{n} γ (A_{j})^{*} γ (A_{j}) + γ (A_{j}) γ (A_{j})^{*} ‖ \leq 4 ‖ γ ‖^{2} ‖ \sum_{j = 1}^{n} A_{j}^{*} A_{j} + A_{j} A_{j}^{*} ‖

for each finite set ${A_{1}, A_{2}, \dots, A_{n}}$ of elements $A_{j}$ of $𝔄$ .