On Fourier algebra homomorphisms

Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism φ : A(G)→B(H) we show that it can be described, in terms of a piecewise affine map α : Y→G with Y in the coset ring of H, as followsφ(f)=f∘αon Y,0off Ywhen G is discrete and amenable. This extends a similar result by Host. We also show that in the same hypothesis the range of a completely bounded algebra homomorphism φ : A(G)→A(H) is as large as it can possibly be and it is equal to a well determined set. The same description of the range is obtained for bounded algebra homomorphisms, this time when G and H are locally compact groups with G abelian.