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.

Theorem.^[1][2] If ${\displaystyle \gamma }$ is a bounded, linear mapping of one C*-algebra ${\displaystyle {\mathfrak {A}}}$ into another C*-algebra ${\displaystyle {\mathfrak {B}}}$, then

${\displaystyle \left\|\sum _{j=1}^{n}\gamma (A_{j})^{*}\gamma (A_{j})+\gamma (A_{j})\gamma (A_{j})^{*}\right\|\leq 4\|\gamma \|^{2}\left\|\sum _{j=1}^{n}A_{j}^{*}A_{j}+A_{j}A_{j}^{*}\right\|}$

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

• Haagerup-Pisier inequality
• Christensen-Haagerup Principle

• Pisier, Gilles (1978), "Grothendieck's theorem for noncommutative C^∗-algebras, with an appendix on Grothendieck's constants", Journal of Functional Analysis, 29 (3): 397–415, doi:10.1016/0022-1236(78)90038-1, MR 0512252.
• Kadison, Richard V. (1993), "On an inequality of Haagerup–Pisier", Journal of Operator Theory, 29 (1): 57–67, MR 1277964.