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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).

46L30 States of selfadjoint operator algebras
46B20 Geometry and structure of normed linear spaces
46L05 General theory of \(C^*\)-algebras
Full Text: DOI
[1] Bergh, J; Löfström, J, Interpolation spaces, () · Zbl 0128.35104
[2] Grothendieck, A, Resumé de la théorie métrique des produits tensorielles topologiques, Bol. soc. mat. sao paolo, 8, 1-79, (1956) · Zbl 0074.32303
[3] Haagerup, U, Solution of the similarity problem for cyclic representations of C∗-algebras, Ann. of math., 118, 215-240, (1983) · Zbl 0543.46033
[4] Heinrich, S, Ultraproducts in Banach space theory, J. reine angew. math., 313, 72-104, (1980) · Zbl 0412.46017
[5] Johnson, B.E; Kadison, R.V; Ringrose, J.R, Cohomology of operator algebras. III. reducting to normal cohomology, Bull. soc. math. France, 100, 73-96, (1972) · Zbl 0234.46066
[6] Kaijser, S, A simple-minded proof of the pisier-Grothendieck inequality, () · Zbl 0547.46038
[7] Pedersen, G.K, C∗-algebras and their automorphism groups, (1979), Academic Press New York · Zbl 0416.46043
[8] Pisier, G, Grothendieck’s theorem for non-commutative C∗-algebras with an appendix on Grothendieck’s constant, J. funct. anal., 29, 397-415, (1978) · Zbl 0388.46043
[9] Szankowski, A, B(H) does not have the approximation property, Acta math., 147, 89-108, (1981) · Zbl 0486.46012
[10] Takesaki, M, Theory of operator algebras I, (1979), Springer-Verlag Berlin · Zbl 0990.46034
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