HOMOMORPHISMS WITH SMALL BOUND BETWEEN FOURIER ALGEBRAS

Inspired by Kalton and Wood's work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier-Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism T between the associated Fourier algebras (resp. Fourier-Stieltjes algebras) with completely bounded norm parallel to T parallel to(cb) < root 3/2 (resp. parallel to T parallel to(cb) < root 5/2). We show similar results involving the norm distortion parallel to T parallel to parallel to T-1 parallel to with universal but non-explicit bound. Our results subsume Walter's well-known structural theorems and also Lau's theorem on the second conjugate of Fourier algebras.