Let G be a locally compact group. Dixmier's unitarization theorem for bounded continuous group representations may be restated as follows: if G is amenable, then every bounded representation for the group algebra on a Hilbert space, pi:L^1(G) --> B(H), admits an invertible S in BB(H) for which

S pi(x) S^{-1} is a *-representation, and ||S|| ||S^{-1}|| <= ||pi||^2. (DAGGER)

In the `90s, Pisier showed that ($\dagger$) implies amenability of G. The Fourier algebra A(G) is the dual object to L^1(G) in a manner which generalizes Pontryagin duality. It is a commutative self-adjoint Banach algebra of functions on G which is the predual of the von Neuman algebra generated by the left regular representation of G. As such, the operator space structure on A(G) is generally non-trivial. However, every *-representation of A(G) factors through the commutative C*-algebra of continuous functions vanishing at infinity C_0(G), and hence is completely bounded. Due to the considerations around the duality of A(G) with L^1(G), we suspect that for any completely bounded representation pi : A(G) --> B(H) that there is an S in B(H) for which an analogue of (DAGGER) holds.

H.H. Lee (Seoul) and E. Samei (Saskatchewan) and I have found a proof for this result for a wide class of groups which includes amenable groups and small-invariant neighbourhood (hence discrete) groups.