Homomorphisms of Fourier algebras and transference results

We prove that if $\rho: A(H) \to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $\alpha : G \to H$, then $\rho$ extends to a w*-w* continuous map between the corresponding $L^\infty$ algebras if and only if $\alpha$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $\alpha$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure of $H$, then $(\alpha\times\alpha)^{-1}$ preserves sets of compact operator synthesis, and conversely when $\alpha$ is onto. We also prove similar preservation properties for operator Ditkin sets and operator M-sets, obtaining preservation properties for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.