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 pi : L-1(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 (a priori unbounded) map pi* : L-#(1)(G)-> B(H) given by pi* (omega) = pi(omega(#))*. We show that the corepresentation V-pi of L-infinity(G) associated to pi is invertible if and only if both pi and pi* are completely bounded. To prove the " if" part of this claim we show that coefficient operators of such representations give rise to completely bounded multipliers of the dual convolution algebra L-#(1) (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, 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 G such that L-1(G) admits bounded, but not completely bounded, representations.