Khintchine type inequalities for reduced free products and Applications

TL;DR

This paper establishes Khintchine type inequalities for reduced free products of $C^*$-algebras and von Neumann algebras, leading to new results on approximation properties and stability of exactness and CCAP.

Contribution

It introduces Khintchine inequalities for fixed-length words in reduced free products and applies them to prove stability of approximation properties.

Findings

Projection onto fixed-length words is completely bounded with norm depending linearly on length d.

Reduced free product of finite-dimensional $C^*$-algebras has the CCAP.

Free product of weakly amenable groups with constant 1 remains weakly amenable.

Abstract

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.