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The Grothendieck inequality for bilinear forms on \(C^*\)-algebras. (English) Zbl 0593.46052
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 \(\phi_ 1\), \(\phi_ 2\) on A and two states \(\psi_ 1\), \(\psi_ 2\) on B, such that \[ | V(x,y)| \leq \| V\| (\phi_ 1(x^*x)+\phi_ 2(xx^*))^{1/2}(\psi_ 1(y^*y)+\psi_ 2(yy\quad^*))^{1/2} \] for all \(x\in A\) and all \(y\in B\). The inequality improves a result of G. Pisier [J. Funct. Anal. 29, 397- 415 (1978; Zbl 0388.46043)], where a similar inequality is proved for bilinear forms satisfying a certain approximability condition. It follows from the inequality, that every bounded linear map from a \(C^*\)-algebra into the dual of a \(C^*\)-algebra has a factorization through a Hilbert space. This result has recently been generalized by G. Pisier to linear maps from a \(C^*\)-algebra into any complex Banach space of cotype 2: [cf. ”Factorization of operators through \(L_{p\infty}\) or \(L_{p1}\) and non-commutative generalizations”, preprint (1986)].
The inequality was in [U. Haagerup: Invent. Math. 74, 305-319 (1983; Zbl 0529.46041)] used to prove, that the class of amenable \(C^*\)-algebras coincides with the class of nuclear \(C^*\)-algebras. (The implication ”amenable \(\Rightarrow nuclear''\) was proved by Connes in 1978).

MSC:
46L30 States of selfadjoint operator algebras
46B20 Geometry and structure of normed linear spaces
46L05 General theory of \(C^*\)-algebras
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