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 $𝔄$ into another C*-algebra $𝔅$, 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 $𝔄$.

See also

• Haagerup-Pisier inequality
• Christensen-Haagerup Principle

References

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