The Grothendieck inequality for bilinear forms on C∗-algebras

Abstract

The following generalization of Grothendieck's inequality is proved: For any bounded bilinear form V on a pair of C∗-algebras A, B, there exist two states ϕ 1, ϕ 2 on A and two states ψ 1, ψ 2 on B, such that |V(x,y)|⩽‖V‖(ϕ 1(x ∗x)+ϕ 2(xx 2)) 1 2 (φ 1(y ∗y)+ϕ 2(yy 2)) 1 2 for all x ϵ A and all y ϵ B. An inequality of this type was proved a few years ago by Pisier in the case where one of the C∗-algebras has the bounded approximation property. It follows from the above inequality that any bounded linear map T of a C∗-algebra into the dual of a C∗-algebra has a factorization T = R ∘ S through a Hilbert space, such that ‖ R ‖ ‖ S ‖ ⩽ 2 ‖ T ‖.