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 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. i 1/2 Mm ,

The Haagerup property for locally compact quantum groups

“…We could also prove the backward implication of the above theorem using Lemma 7.25 and the equivalence of conditions (i) and (iii) in Theorem 7.18. Indeed, let [35] that weak amenability (with the Cowling-Haagerup constant equal 1) is preserved under taking free products of discrete quantum groups, extending a result of E. Ricard and Q. Xu for discrete groups [66].…”

Weak amenability is stable under graph products

“…We can now sketch the proof of the next theorem, inspired by Ricard and Xu's proof in [23], Proof sketch. If A and B are weakly amenable, then we have two sequences consisting of finitely supported completely bounded functions {f n A } and {f n B } which go to the identity and whose cb-norms go to one as n goes to infinity.…”

Lacunary Fourier Series for Compact Quantum Groups

“…Let K 1 be the Sidon constant for E and K 2 that for F. By Remark 3.2, we may assume that 1 / ∈ E and 1 / ∈ F. Now for any x ∈ Pol E (G 1 * G 2 ) and y ∈ Pol F (G 1 * G 2 ), we have h 1 (x) = 0, h 2 (y) = 0, and x and y are free. Then it is well known and easy to see from the construction of reduced free products that max{ x ∞ , y ∞ } ≤ x + y ∞ (see [Voi98,Jun05,RX06] for more information on the norm estimates related to freeness). Hence…”