Éric Ricard

Khintchine type inequalities for reduced free products and applications

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 Dykemas 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.

Higher order extension of Lowner's theory: Operator k-tone functions

The new notion of operator/matrix k-tone functions is introduced, which is a higher order extension of operator/matrix monotone and convex functions. Differential properties of matrix k-tone functions are shown. Characterizations, properties, and examples of operator k-tone functions are presented. In particular, integral representations of operator k-tone functions are given, generalizing familiar representations of operator monotone and convex functions.