Contractive homomorphisms of the Fourier algebras

Let G and H be locally compact groups. We prove that, for every contractive homomorphism θ from A( G ), the Fourier algebra of G , into B( H ), the Fourier–Stieltjes algebra of H , there exist a continuous group homomorphism or anti‐homomorphism τ from an open subgroup Ω of H into G and elements u 0 ∈ G , r 0 ∈ H such that θ ( f ) ( t ) = {f ( u 0 τ ( r 0 τ ) )if   t   ∊   r 0 − 1 Ω ,0if   t   ∊   H \ r 0 − 1 Ω .We also characterize positive homomorphisms from A( G ) into B( H ) and contractive isomorphisms from B( G ) onto B( H ).