Completely bounded representations of convolution algebras of locally compact quantum groups

Given a locally compact quantum group G, we study the structure of completely bounded homomorphisms π:L1(G)→B(H), and the question of when they are similar to ∗-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra A(G)), we are led to consider the associated map π∗:L1♯(G)→B(H) given by π∗(ω)=π(ω♯)∗. We show that the corepresentation Vπ of L∞(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 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 G such that L1(G) admits bounded, but not completely bounded, representations.