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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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##### References:
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