Non-unitarizable representations of compact and discrete quantum groups

Abstract

We study the similarity problem for representations of compact and discrete quantum groups. We first prove that every non-degenerate contractive representation of a compact or discrete quantum group is automatically unitary. Then, for compact quantum groups, we develop an interpolation method for constructing explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$. In particular, this applies to all non-Kac compact quantum groups whose dual has subexponential growth, including the Drinfeld--Jimbo quantum groups $G_q$, as well as to the non-Kac free unitary quantum groups $U_F^{+}$ with $F\in {\mathrm{GL}}_2(\mathbb{C})$. On the discrete quantum group side, we establish a quantum analogue of the classical lifting principle for non-unitarizable uniformly bounded representations. Combining this lifting principle with recent results on maximal Kac quantum subgroups, we construct explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$ for all unitary free quantum groups $\mathbb{F}{U}_F$ and all non-amenable orthogonal free quantum groups $\mathbb{F}{O}_F$.