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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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