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 : Y ⊂ H → G (where Y is an element of the open coset ring) induces a completely bounded homomorphism : A(G) → B(H) by setting u = u ◦ on Y and u = 0 off of Y. We show that if G is amenable then any completely bounded homomorphism : 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. © 2005 Elsevier Inc. All rights reserved.