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

$\Vert \sum _{j=1}^n\gamma (A_j{)}^{*}\gamma (A_j)+\gamma (A_j)\gamma (A_j{)}^{*}\Vert \le 4\Vert \gamma {\Vert }^2\Vert \sum _{j=1}^n{A}_j^{*}{A}_j+A_j{A}_j^{*}\Vert$

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

See also

• Haagerup-Pisier inequality
• Christensen-Haagerup Principle

References

• "Grothendieck's theorem for noncommutative C^∗-algebras, with an appendix on Grothendieck's constants", Journal of Functional Analysis 29 (3): 397–415, 1978, doi:10.1016/0022-1236(78)90038-1 .
• "On an inequality of Haagerup–Pisier", Journal of Operator Theory 29 (1): 57–67, 1993, http://www.theta.ro/jot/archive/1993-029-001/1993-029-001-004.html .