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 $π:L^1(\mathbb G)\rightarrow\mathcal B(H)$, 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 $π^*:L^1_\sharp(\mathbb G) \rightarrow \mathcal B(H)$ given by $π^*(ω) = π(ω^\sharp)^*$. We show that the corepresentation $V_π$ of $L^\infty(\mathbb G)$ associated to $π$ is invertible if and only if both $π$ and $π^*$ 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, $π$ is similar to a *-homomorphism if and only if $π^*$ is completely bounded. For compact Kac algebras, and for certain cases of $A(G)$, we show that any completely bounded homomorphism $π$ is similar to a *-homomorphism, without further assumption on $π^*$. 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.

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