Pisier–Ringrose inequality

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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 [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

Notes

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

References