Cowling–Haagerup constant of the product of discrete quantum groups

Abstract

We show that (central) Cowling–Haagerup constant of discrete quantum groups is multiplicative \Lambda_{\mathrm{cb}}({\mathrm I\mathrlap{\! \Gamma}\,\,\,}_{1}\times {\mathrm I\mathrlap{\! \Gamma}\,\,\,}_{2}) =\Lambda_{\mathrm{cb}}({\mathrm I\mathrlap{\! \Gamma}\,\,\,}_1)\, \Lambda_{\mathrm{cb}}({\mathrm I\mathrlap{\! \Gamma}\,\,\,}_{2}) , which extends the result of Freslon (2015) to general (not necessarily unimodular) discrete quantum groups. The crucial feature of our approach is considering algebras \mathrm{C}(\hat{\mathrm I\mathrlap{\! \Gamma}\,\,\,}), \operatorname{L}^{\infty}(\hat{\mathrm I\mathrlap{\! \Gamma}\,\,\,}) as operator modules over \operatorname{L}^{1}(\hat{\mathrm I\mathrlap{\! \Gamma}\,\,\,}) .