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 $\gamma$ is a bounded, linear mapping of one C*-algebra $\mathfrak{A}$ into another C*-algebra $\mathfrak{B}$, then

$\left\|\sum_{j=1}^n \gamma(A_j)^* \gamma(A_j) + \gamma(A_j) \gamma(A_j)^*\right\| \le 4 \|\gamma \|^2 \left\| \sum_{j=1}^n A_j^*A_j + A_j A_j^* \right\|$

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

See Also

• Haagerup-Pisier inequality
• Christensen-Haagerup Principle

Notes

1. Kadison (1993), Theorem D, p. 60.
2. Pisier (1978), Corollary 2.3, p. 410.

References

• 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 512252.
• Kadison, Richard V. (1993), "On an inequality of Haagerup–Pisier", Journal of Operator Theory 29 (1): 57–67, MR 1277964.