The Grothendieck inequality for bilinear forms on C∗-algebras
The Grothendieck inequality for bilinear forms on C∗-algebras
复制标题
C*-代数双线性形式的格洛滕迪克不等式
DOI：
10.1016/0001-8708(85)90026-x
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发表时间：
1985
影响因子：
1.7
通讯作者：
U. Haagerup
中科院分区：
文献类型：
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作者：
U. Haagerup
证明了格罗thendieck不等式的以下推广：对于C∗-代数对a， B上的任何有界双线性形式V，在a上存在两个态φ 1， φ 2和在B上存在两个态ψ 1， ψ 2，使得| V (x, y)|≥‖V‖(φ 1 (x∗x)+ φ 2 (xx 2)) 1 2 (φ 1 (y∗y)+ φ 2 (yy 2)) 1 2对于所有xϵA和所有yϵB。几年前，Pisier在一个C * -代数具有有界近似性质的情况下证明了这种类型的不等式。由上述不等式可以得出，任何C∗-代数的有界线性映射T到C∗-代数的对偶都有因式分解T= R°S，通过希尔伯特空间，使得‖R‖‖S‖≥2‖T‖。
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 xϵA and all yϵB. 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‖.