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 ${\displaystyle \gamma }$ 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 ${\displaystyle {\mathfrak {A}}}$.

See also

• Haagerup-Pisier inequality
• Christensen-Haagerup Principle