Completely bounded representations of convolution algebras of locally compact quantum groups

Given a locally compact quantum group $\mathbb G$, we study the structure of completely bounded homomorphisms $\pi:L^1(\mathbb G)\rightarrow\mathcal B(H)$, and the question of when they are similar to $\ast$-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra $A(G)$), we are led to consider the associated map $\pi^*:L^1_\sharp(\mathbb G) \rightarrow \mathcal B(H)$ given by $\pi^*(\omega) = \pi(\omega^\sharp)^*$. We show that the corepresentation $V_\pi$ of $L^\infty(\mathbb G)$ associated to $\pi$ is invertible if and only if both $\pi$ and $\pi^*$ are completely bounded. Moreover, we show that the co-efficient operators of such representations give rise to completely bounded multipliers of the dual convolution algebra $L^1(\hat \mathbb G)$. An application of these results is that any (co)isometric corepresentation is automatically unitary. An averaging argument then shows that when $\mathbb G$ is amenable, $\pi$ is similar to a *-homomorphism if and only if $\pi^*$ is completely bounded. For compact Kac algebras, and for certain cases of $A(G)$, we show that any completely bounded homomorphism $\pi$ is similar to a *-homomorphism, without further assumption on $\pi^*$. Using free product techniques, we construct new examples of compact quantum groups $\mathbb G$ such that $L^1(\mathbb G)$ admits bounded, but not completely bounded, representations.