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 [math]\displaystyle{ \gamma }[/math] is a bounded, linear mapping of one C*-algebra [math]\displaystyle{ \mathfrak{A} }[/math] into another C*-algebra [math]\displaystyle{ \mathfrak{B} }[/math], then

[math]\displaystyle{ \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\| }[/math]

for each finite set [math]\displaystyle{ \{ A_1, A_2, \ldots, A_n \} }[/math] of elements [math]\displaystyle{ A_j }[/math] of [math]\displaystyle{ \mathfrak{A} }[/math].

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 .