关于群同态的合成与迁移性质
On synthetic and transference properties of group homomorphisms
摘要：
本文研究了任意局部紧第二可数群$G$和$H$的Borel同态$θ: G\rightarrow H$，其中测度$$θ_*(μ)(α)=μ(θ^{-1}(α))\quad \text{for } \quad α\subseteq H $$关于$ν,$是绝对连续的，其中$μ$（resp. $ν$）是$G,$的Haar度量（分别为）。$H$）。本文从极大交换自伴代数双模类中定义了一个自然映射$\mathcal G$将$B(L^2(H))$中的masa双模（masa bimodules）引入到$B(L^2(G))$中的masa bimodules类中，并利用它证明了：如果$k\subseteq G\times G$是算子综合集，则$(θ\times θ)^{-1} (k)$也是算子综合集，如果$E\subseteq H$是Fourier代数$A(H)$的局部综合集，则$θ^{-1}(E)\subseteq G$是$A(G).$的一个局部综合集。$M_1$ -set），则$E$是$M$ -set（相应地，$M_1$ -集合），并且如果$Bim(I^\bot )$是由理想$I$在$VN(G)$中的零化子生成的masa双模，则存在理想$J$使得$\mathcal G(Bim(I^\bot ))=Bim(J^\bot ).$如果这个理想$J$是重数理想，则$I$是重数理想。如果$θ_*(μ)$是$θ(G)$的Haar度量，我们证明$J$等于$ρ(I),$生成的理想$ρ_*(I)$，其中$ρ(u)=u\circ θ, \;\;\forall \;u\;\in \;I.$
英文摘要：
We study Borel homomorphisms $θ: G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$θ_*(μ)(α)=μ(θ^{-1}(α))\quad \text{for } \quad α\subseteq H $$ is absolutely continuous with respect to $ν,$ where $μ$ (resp. $ν$) 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 $(θ\times θ)^{-1} (k)$ 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 $θ^{-1}(E)\subseteq G$ is a set of local synthesis for $A(G).$ We also prove that if $θ^{-1}(E)$ 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 $θ_*(μ)$ is a Haar measure for $θ(G)$ we show that $J$ is equal to the ideal $ρ_*(I)$ generated by $ρ(I),$ where $ρ(u)=u\circ θ, \;\;\forall \;u\;\in \;I.$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
