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Bounded linear operators between $$C^*$$-algebras. (English) Zbl 0803.46064
The authors prove “higher degree” versions of Grothendieck inequalities for bounded linear operators between $$C^*$$-algebras which in degree one were first conjectured by Ringrose and established by the second named author [J. Funct. Anal. 29, 397-415 (1978; Zbl 0388.46043)]. These are applied to compute the norm of random series with coefficients in a noncommutative $$L_ 1$$-space, and to elucidate the relation between completely bounded linear projections from $$B(H)$$ onto a subspace $$S$$ and the bounded linear projections from $$B(\ell_ 2)\otimes B(H)$$ onto $$B(\ell_ 2)\otimes S$$. If $$S$$ is weak-$$*$$-closed a completely bounded linear projection from $$B(H)$$ onto $$S$$ is shown to exist iff there is a bounded linear projection from $$B(\ell_ 2\otimes H)$$ onto the weak-$$*$$- closure $$B(\ell_ 2)\overline{\otimes} S$$ of $$B(\ell_ 2)\otimes S$$ in $$B(\ell_ 2\otimes H)$$. In particular, they prove that this is not the case for the von Neumann algebra $$VN(F_ n)$$ associated to the free group of $$n\geq 2$$ generators. This implies that $$VN(F_ n)$$ is not a complemented subspace of $$B(H)$$.

##### MSC:
 46L05 General theory of $$C^*$$-algebras 46L10 General theory of von Neumann algebras 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras
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