论文标题
关于组同态的合成和转移特性
On synthetic and transference properties of group homomorphisms
论文作者
论文摘要
我们研究了borel同构$θ：g \ rightarrow h $，用于任意局部紧凑的第二个可计数组$ g $和$ h $，$$θ_*（μ）（α）=μ（α）=μ（θ^{ - 1}（-1}（α）（α））\ quad text { $μ$（分别$ν$）是$ g的HAAR措施，$ $（分别$ h $）。 We define a natural mapping $\mathcal G$ from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules) in $B(L^2(H))$ into the class of masa bimodules in $B(L^2(G))$ and we use it to prove that if $k\subseteq G\times G$ is a set of operator synthesis, then $(θ\times θ)^{-1} (k)$ is also a set of operator synthesis and if $E\subseteq H$ is a set of local synthesis for the Fourier algebra $A(H)$, then $θ^{-1}(E)\subseteq G$ is a set of local synthesis for $A(G).$ We also prove that if $θ^{ - 1}（e）$是$ m $ - set（分别为$ $ m_1 $ - set），然后$ e $是$ m $ - set（resp。$ $ m_1 $ - set），如果$ bim（i^\ bot）$是Masa bimodule，则是$ i $ $ i $ $ j $ j $ j $ j $ j $ vn（g）的$ i $ j），然后g（bim（i^\ bot））= bim（j^\ bot）。$如果此理想$ j $是多重性的理想，那么$ i $是多重性的理想。如果$θ_*（μ）$是$θ（g）$的HAAR度量，我们表明$ j $等于$ρ（i）生成的理想$ρ_*（i）$，其中$ρ（i），$ρ（u）= u \ circtim = u \ circtim;
We study Borel homomorphisms $θ: G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$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 $ρ(u)=u\circ θ, \;\;\forall \;u\;\in \;I.$
