[Paper Review] Khintchine type inequalities for reduced free products and Applications

Éric Ricard, Quanhua Xu|arXiv (Cornell University)|May 14, 2005

Advanced Operator Algebra Research23 references3 citations

TL;DR

This paper establishes Khintchine-type inequalities for words of fixed length in reduced free products of C*-algebras and von Neumann algebras, proving that the projection onto the subspace of words of length $d$ is completely bounded with norm linear in $d$. These results are applied to show stability of exactness and the completely contractive approximation property (CCAP) under reduced free products, including proving that free products of weakly amenable groups with constant 1 remain weakly amenable with constant 1.

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.

Motivation & Objective

• To extend Khintchine-type inequalities to words of fixed length $d$ in reduced free products of $C^*$-algebras and von Neumann algebras.
• To establish that the projection onto the subspace of words of length $d$ is completely bounded with norm linear in $d$.
• To apply these inequalities to prove stability of approximation properties—especially exactness and the completely contractive approximation property (CCAP)—under reduced free products.
• To resolve open questions on the preservation of weak amenability and CCAP in free products of group $C^*$-algebras and injective von Neumann algebras.

Proposed method

• Prove a free product version of Buchholz’s inequality with constant $2d+1$ for words of length $d$ in reduced free products of $C^*$-algebras.
• Use operator space theory, particularly the Haagerup tensor product and row/column Hilbertian spaces, to analyze the structure of the free product.
• Apply the Khintchine-type inequalities to show that the projection onto the $d$-th homogeneous component is completely bounded with norm $O(d)$.
• Utilize the universal property of the modular Haagerup tensor product to extend inequalities to the amalgamated case.
• Establish that exactness is preserved under reduced free products by proving 1-exactness of relevant Haagerup tensor products.
• Leverage the linear dependence on $d$ to prove stability of CCAP for finite-dimensional $C^*$-algebras and weak-* CCAP for injective von Neumann algebras.

Experimental results

Research questions

• RQ1 Can Khintchine-type inequalities be extended from length 1 to arbitrary fixed length $d$ in reduced free products of $C^*$-algebras?
• RQ2 Is the projection onto the subspace of words of length $d$ completely bounded, and what is the dependence of its norm on $d$?
• RQ3 Does the completely contractive approximation property (CCAP) persist under reduced free products of $C^*$-algebras and von Neumann algebras?
• RQ4 Is weak amenability preserved under free products of discrete groups with constant 1?
• RQ5 Can the stability of exactness under reduced free products be proven via operator space techniques and Khintchine inequalities?

Key findings

• The projection onto the subspace generated by words of fixed length $d$ in a reduced free product is completely bounded with norm at most $2d+1$, linear in $d$.
• The free product version of Haagerup-Pisier’s inequality is extended to arbitrary length $d$, with the upper bound constant $2d+1$.
• A reduced free product of finite-dimensional $C^*$-algebras has the completely contractive approximation property (CCAP).
• A von Neumann reduced free product of injective von Neumann algebras with respect to normal states has the weak-* CCAP.
• A free product of weakly amenable discrete groups with constant 1 remains weakly amenable with constant 1.
• The stability of exactness under reduced free products is established via Khintchine inequalities and 1-exactness of Haagerup tensor products.