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\|\leq 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

↑ Kadison (1993), Theorem D, p. 60.
↑ 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 .

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