The Grothendieck inequality for bilinear forms on \(C^*\)-algebras.

*(English)*Zbl 0593.46052The 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).

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 |

##### Keywords:

Grothendieck’s inequality; states; bounded linear map from a \(C^*\)- algebra into the dual of a; \(C^*\)-algebra; factorization through a Hilbert space; amenable \(C^*\)-algebras; nuclear \(C^*\)-algebras; bounded linear map from a \(C^*\)-algebra into the dual of a \(C^*\)- algebra
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##### References:

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