On synthetic and transference properties of group homomorphisms

TL;DR

This paper investigates the properties of Borel group homomorphisms between locally compact second countable groups, focusing on measure-theoretic, operator synthesis, and ideal multiplicity aspects, establishing new connections and invariance results.

Contribution

It introduces a natural mapping between masa bimodules and proves preservation of operator and local synthesis properties under inverse images of group homomorphisms.

Findings

Inverse images of sets of operator synthesis are also sets of operator synthesis.

Inverse images of sets of local synthesis are also sets of local synthesis.

Ideals of multiplicity are preserved under the natural mapping induced by the homomorphism.

Abstract

We study Borel homomorphisms $\theta : G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$\theta_*(\mu )(\alpha )=\mu (\theta ^{-1}(\alpha ))\quad \text{for } \quad \alpha \subseteq H $$ 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 $Bim(I^\bot )$ is the masa bimodule generated by the annihilator of the ideal $I$ in $VN(G)$, then there exists an ideal $J$ such that $\mathcal G(Bim(I^\bot ))=Bim(J^\bot ).$ If this ideal $J$ is an ideal of multiplicity then $I$ is an ideal of multiplicity. In case $\theta_*(\mu )$ is a Haar measure for $\theta (G)$ we show that $J$ is equal to the ideal $\rho_*(I)$ generated by $\rho (I),$ where $\rho (u)=u\circ \theta , \;\;\forall \;u\;\in \;I.$