[Paper Review] A note on weak amenability for reduced free products of discrete quantum groups

Amaury Freslon|arXiv (Cornell University)|Nov 22, 2011

Advanced Operator Algebra Research21 references3 citations

TL;DR

This paper establishes that the Cowling-Haagerup constant of a reduced free product of discrete quantum groups remains equal to 1 if each factor has this constant. Using a quantum generalization of the Khintchine inequality and Daws' characterization of completely bounded multipliers, the authors construct a unital completely positive approximation in the completely bounded operator norm, proving weak amenability is preserved under free products in this setting.

ABSTRACT

We prove that the Cowling-Haagerup constant of a reduced free product of weakly amenable discrete quantum groups with Cowling-Haagerup constant equal to 1 is again equal to 1.

Motivation & Objective

• To extend Ricard and Xu's result on weak amenability in free products of discrete groups to the setting of discrete quantum groups.
• To establish that the Cowling-Haagerup constant is preserved under reduced free products when all factors have constant 1.
• To leverage Daws' characterization of completely bounded multipliers on quantum groups to construct norm-approximating sequences.
• To prove that the reduced free product of weakly amenable discrete quantum groups with Cowling-Haagerup constant 1 is itself weakly amenable with the same constant.

Proposed method

• Utilizes Daws' characterization of completely bounded multipliers on discrete quantum groups via Hilbert space-valued maps α and β.
• Constructs a net of finitely supported elements in ℓ∞(Ĝi) converging pointwise to the identity with cb-norm approaching 1.
• Applies a quantum version of the Khintchine inequality to control the norm of the multiplier approximation.
• Uses unital completely positive maps Mγ̃ to approximate the multiplier ma in both the standard and opposite L² norms.
• Verifies τ-invariance of the approximating maps to ensure compatibility with the trace on the von Neumann algebra.
• Applies Theorem 4.1 to conclude that the Cowling-Haagerup constant of the free product is 1.

Experimental results

Research questions

• RQ1 Does the Cowling-Haagerup constant remain 1 under reduced free products of discrete quantum groups when each factor has constant 1?
• RQ2 Can the classical proof technique of Ricard and Xu for groups be generalized to the non-commutative setting of discrete quantum groups?
• RQ3 Is there a quantum analogue of the Khintchine inequality that allows control over the completely bounded norm of multipliers in free products?
• RQ4 Can unital completely positive approximations be constructed in the completely bounded operator norm and L² norms for quantum group multipliers?
• RQ5 Does the τ-invariance of the approximating maps ensure compatibility with the trace on the von Neumann algebra of the free product?

Key findings

• The Cowling-Haagerup constant of the reduced free product of discrete quantum groups is equal to 1 if each factor has constant 1.
• The proof relies on constructing a unital completely positive approximation to the multiplier in the completely bounded operator norm.
• The approximation is simultaneously valid in both the standard and opposite L² norms, with error bounded by 6ε.
• The approximating maps Mγ̃ are shown to be τ-preserving, ensuring compatibility with the trace on the von Neumann algebra.
• The result extends Ricard and Xu’s theorem on groups to the setting of discrete quantum groups via quantum group duality and Daws’ multiplier characterization.
• Examples include free orthogonal and unitary quantum groups, whose duals have Cowling-Haagerup constant 1, and their free products inherit this property.