Cowling–Haagerup constant of the product of discrete quantum groups

Abstract

We show that (central) Cowling–Haagerup constant of discrete quantum groups is multiplicative ${\Lambda }_{\mathrm{cb}}({ℾ}_1\times {ℾ}_2)={\Lambda }_{\mathrm{cb}}({ℾ}_1){\Lambda }_{\mathrm{cb}}({ℾ}_2)$, which extends the result of Freslon (2015) to general (not necessarily unimodular) discrete quantum groups. The crucial feature of our approach is considering algebras $\mathrm{C}(\widehat{ℾ}),L^{\infty }(\widehat{ℾ})$ as operator modules over $L^1(\widehat{ℾ})$.