论文标题
边界图，细菌和准常规表示
Boundary maps, germs and quasi-regular representations
论文作者
论文摘要
我们研究了$ c^*$ - 边界行动稳定器的准常规表示的代数的奇特和理想结构。我们的主要工具是边界图的概念，即$γ$ Equivariant Unital完全正面地图，从$γ$ - $ C^*$ - 代数到$ c（\partial_fγ）$，其中$ \partial_fγ$表示$γ$的furstenberg边界。对于统一表示$π$来自边界动作的细菌的群体，我们表明$ c^*_π（γ）$上有独特的边界图。因此，我们完全描述了$ c^*$ - 代数$ c^*_π（γ）$，以及对于任何$γ$ cub-coundary $ x $，我们完全表征了$ c^*$ - 代数的简单性，由Quasi-Rectular表示$λ_$λ/γ_x$ gots $γ_x} $ cobs y notem uspot not uspoct $λ_{ $ x \ in x $。作为一个应用程序，我们表明由quasi-regular表示产生的$ c^*$ - 代数$λ_{t/f} $与汤普森的组相关的$ f \ leq t $不承认迹线，而且很简单。
We investigate the tracial and ideal structures of $C^*$-algebras of quasi-regular representations of stabilizers of boundary actions. Our main tool is the notion of boundary maps, namely $Γ$-equivariant unital completely positive maps from $Γ$-$C^*$-algebras to $C(\partial_FΓ)$, where $\partial_FΓ$ denotes the Furstenberg boundary of a group $Γ$. For a unitary representation $π$ coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on $C^*_π(Γ)$. Consequently, we completely describe the tracial structure of the $C^*$-algebras $C^*_π(Γ)$, and for any $Γ$-boundary $X$, we completely characterize the simplicity of the $C^*$-algebras generated by the quasi-regular representations $λ_{Γ/Γ_x}$ associated to stabilizer subgroups $Γ_x$ for any $x\in X$. As an application, we show that the $C^*$-algebra generated by the quasi-regular representation $λ_{T/F}$ associated to Thompson's groups $F\leq T$ does not admit traces and is simple.
