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Khintchine type inequalities for reduced free products and applications. (English) Zbl 1170.46052

Summary: We prove Khintchine type inequalities for words of a fixed length in a reduced free product of \(C^*\)-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length \(d\) is completely bounded with norm depending linearly on \(d\). We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema’s theorem on the stability of exactness under the reduced free product for \(C^*\)-algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite-dimensional \(C^*\)-algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak\(^*\)-CCAP. In the case of group \(C^*\)-algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.

MSC:

46L09 Free products of \(C^*\)-algebras
46L35 Classifications of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
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