Khintchine type inequalities for reduced free products and Applications

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.