Completely bounded homomorphisms of the Fourier algebras

For locally compact groups G and H let A(G) denote the Fourier algebra of G and B(H) the Fourier-Stieltjes algebra of H. Any continuous piecewise affine map α:Y⊂H→G (where Y is an element of the open coset ring) induces a completely bounded homomorphism Φα:A(G)→B(H) by setting Φαu=u∘α on Y and Φαu=0 off of Y. We show that if G is amenable then any completely bounded homomorphism Φ:A(G)→B(H) is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of Cohen (Amer. J. Math. 82 (1960) 213-226), Host (Bull. Soc. Math. France (1986) 114) and of the first author (J. Funct. Anal. (2004) 213). We also obtain a description of all the idempotents in the Fourier-Stieltjes algebras which are contractive or positive definite.