Homomorphisms between Fourier algebras

The Fourier algebra $A(G)$ and the Fourier-Stieltjes algebra $B(G)$ of a locally compact group $G$ were introduced by Eymard [1964]. If $G$ is abelian, $A(G)$ and $B(G)$ can be identified, via Fourier transform, with $L^1(\widehat{G})$ and the measure algebra $M(\widehat{G})$ of the dual group $\widehat{G}$, respectively. Cohen [1960] characterized the homomorphisms from $A(H)$ into $B(G)$ for $H$ and $G$ locally compact abelian groups using a characterization of idempotents in $B(G)$. Homomorphisms of Fourier algebras for general locally compact groups were studied by Ilie-Spronk [2005] and Daws [2022].
We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norm in the Fourier algebra $A(G)$ and the Fourier-Stieltjes algebra $B(G)$ of a locally compact group $G$. We prove that the existence of idempotents of arbitrarily large norm in $B(G)$ implies the existence of homomorphisms of arbitrarily large norm from $A(H)$ into $B(G)$ for every locally compact group $H$.