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

‖ ∑ j = 1 n γ ( A j ) ∗ γ ( A j ) + γ ( A j ) γ ( A j ) ∗ ‖ ≤ 4 ‖ γ ‖ 2 ‖ ∑ j = 1 n A j ∗ A j + A j A j ∗ ‖ {\displaystyle \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 , … , A n } {\displaystyle \{A_{1},A_{2},\ldots ,A_{n}\}} of elements A j {\displaystyle A_{j}} of A {\displaystyle {\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 0512252.
  • Kadison, Richard V. (1993), "On an inequality of Haagerup–Pisier", Journal of Operator Theory, 29 (1): 57–67, MR 1277964.