Quasi-regular representations of discrete groups and associated $C^*$-algebras
Quasi-regular representations of discrete groups and associated $C^*$-algebras
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DOI：
10.1090/tran/7969
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发表时间：
2019-03
影响因子：
1.3
通讯作者：
Bachir Bekka;Mehrdad Kalantar
中科院分区：
文献类型：
--
作者：
Bachir Bekka;Mehrdad Kalantar
令 $G$ 为可数群。我们在 $G$ 的子群集合 ${\rm Sub}(G)$ 上引入几个等价关系，由与 $H\in {\rm Sub}(G)$ 相关的拟正则表示 $\lambda_{G/H}$ 的属性定义，并将它们与子群的 $G$ 共轭关系进行比较。我们定义一类 ${\rm Sub}_{\rm sg}(G)$ 子群（这些子群具有一定的谱间隙属性），并表明它们是刚性的，即上述任何一个等价关系的 $H\in {\rm Sub}_{\rm sg}(G)$ 的等价类与 $H$ 的 $G$ 共轭类一致。接下来，我们引入第二类 ${\rm Sub}_{\rm w-par}(G)$ 子群（这些子群在某种意义上是弱抛物线的），并建立关于由 $\lambda_{G/H}$ 为子群 $H$ 生成的 $C^*$-代数 $C^*_{\lambda_{G/H}}(G)$ 理想结构的结果，子群 $H$ 属于 ${\rm 类之一子}_{\rm w-par}(G)$ 和 ${\rm Sub}_{\rm sg}(G)$。更一般地，我们的结果对于归纳表示 ${\rm Ind}_H^G \sigma$ 是有效的，其中 $\sigma$ 是 $H\in {\rm Sub}(G)$ 的表示。
Let $G$ be a countable group. We introduce several equivalence relations on the set ${\rm Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations $\lambda_{G/H}$ associated to $H\in {\rm Sub}(G)$ and compare them to the relation of $G$-conjugacy of subgroups. We define a class ${\rm Sub}_{\rm sg}(G)$ of subgroups (these are subgroups with a certain spectral gap property) and show that they are rigid, in the sense that the equivalence class of $H\in {\rm Sub}_{\rm sg}(G)$ for any one of the above equivalence relations coincides with the $G$-conjugacy class of $H$. Next, we introduce a second class ${\rm Sub}_{\rm w-par}(G)$ of subgroups (these are subgroups which are weakly parabolic in some sense) and we establish results concerning the ideal structure of the $C^*$-algebra $C^*_{\lambda_{G/H}}(G)$ generated by $\lambda_{G/H}$ for subgroups $H$ which belong to either one of the classes ${\rm Sub}_{\rm w-par}(G)$ and ${\rm Sub}_{\rm sg}(G)$. Our results are valid, more generally, for induced representations ${\rm Ind}_H^G \sigma$, where $\sigma$ is a representation of $H\in {\rm Sub}(G)$.