Cowling-Haagerup constant of the product of discrete quantum groups
Jacek Krajczok
TL;DR
This work addresses the multiplicativity of the Cowling-Haagerup constant for discrete quantum groups, including non-unimodular cases. It introduces a module-theoretic framework by treating $C(\widehat{\mathbb{L}})$ and $L^\infty(\widehat{\mathbb{L}})$ as $L^1(\widehat{\mathbb{L}})$-modules and defines module variants of the Cowling-Haagerup constants, proving their equality with the classical constants. The main technical achievement is a Brown-Ozawa–type argument adapted to module maps that shows multiplicativity of the module constant for products, which then yields the multiplicativity of both the standard and central Cowling-Haagerup constants for products of discrete quantum groups. Consequently, for discrete quantum groups $\mathbb{L}_1$ and $\mathbb{L}_2$, one has $\Lambda_{cb}(\mathbb{L}_1\times\mathbb{L}_2)=\Lambda_{cb}(\mathbb{L}_1)\Lambda_{cb}(\mathbb{L}_2)$ and $\mathcal{Z}\Lambda_{cb}(\mathbb{L}_1\times\mathbb{L}_2)=\mathcal{Z}\Lambda_{cb}(\mathbb{L}_1)\mathcal{Z}\Lambda_{cb}(\mathbb{L}_2)$, with extensions to infinite direct sums and explicit examples illustrating sharpness. This advances understanding of weak amenability in the quantum setting and clarifies how approximation properties transfer to product structures in operator-algebraic frameworks.
Abstract
We show that (central) Cowling-Haagerup constant of discrete quantum groups is multiplicative, which extends the result of Freslon to general (not necesarilly unimodular) discrete quantum groups. The crucial feature of our approach is considering algebras $\mathrm{C}(\mathbb{G}), \operatorname{L}^{\infty}(\mathbb{G})$ as operator modules over $\operatorname{L}^1(\mathbb{G})$.