Pisier–Ringrose inequality explained

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 ak{A} into another C*-algebra ak{B} , then \left\|\sum ^{n} _{j =1} \gamma(A ^{*} _{j)} \gamma(A _{j)} + \gamma(A _{j)} \gamma(A ^{*\right\|} _{j)} \le 4 \|\gamma \| ^{2} \left\| \sum ^{n} _{j =1} A ^{*A} _{j} + A _{j} A ^{*} _{j} \right\| for each finite set \{ A _{1,} A _{2,} \ldots, A _{n} \} of elements A _{j} of ak{A} .

See also

• Haagerup-Pisier inequality
• Christensen-Haagerup Principle

References

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Notes and References

1. , Theorem D, p. 60.
2. , Corollary 2.3, p. 410.