On synthetic and transference properties of group homomorphisms

Abstract

We study Borel homomorphisms $\theta :G\to H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $${\theta }_{\ast }(\mu )(\alpha )=\mu ({\theta }^{-1}(\alpha ))\text{for }\alpha \subseteq H$$ is absolutely continuous with respect to $\nu ,$ where $\mu$ (resp. $\nu$) is a Haar measure for $G,$ (resp. $H$). We define a natural mapping $\mathcal{G}$ from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules) in $B(L^2(H))$ into the class of masa bimodules in $B(L^2(G))$ and we use it to prove that if $k\subseteq G\times G$ is a set of operator synthesis, then $(\theta \times \theta {)}^{-1}(k)$ is also a set of operator synthesis and if $E\subseteq H$ is a set of local synthesis for the Fourier algebra $A(H)$, then ${\theta }^{-1}(E)\subseteq G$ is a set of local synthesis for $A(G).$ We also prove that if ${\theta }^{-1}(E)$ is an $M$-set (resp. $M_1$-set), then $E$ is an $M$-set (resp. $M_1$-set) and if $Bim(I^{\perp })$ is the masa bimodule generated by the annihilator of the ideal $I$ in $VN(G)$, then there exists an ideal $J$ such that $\mathcal{G}(Bim(I^{\perp }))=Bim(J^{\perp }).$ If this ideal $J$ is an ideal of multiplicity then $I$ is an ideal of multiplicity. In case ${\theta }_{\ast }(\mu )$ is a Haar measure for $\theta (G)$ we show that $J$ is equal to the ideal ${\rho }_{\ast }(I)$ generated by $\rho (I),$ where $\rho (u)=u\circ \theta ,\forall u\in I.$