论文标题
痕迹和c* - 简化的独特性超出了常规表示
Uniqueness of trace and C*-simplicity beyond regular representation
论文作者
论文摘要
如果c*-algebra $c_λ^*（γ）$由左常规表示范围$ \ ell^2（γ）$生成的c*-algebra $c_λ^*（γ）$很简单，则一个离散的组$γ$是c*-simple。在这种情况下，$γ$忠实地在furstenberg边界$ \partial_fγ$上行动，并且在$C_λ^*（γ）$上有一个独特的痕迹。在本文中，我们研究了C*-Algebra $C_π^*（γ）$的独特跟踪特性，该$由任意单位表示$π：γ\ to B（h_π）$产生，并将其与Furstenberg-Hamana边界$ \ Mathcal b _ Mathcal b _ Mathcal b_ p _ $ \ us的忠诚相关联。在$c_π^*（γ）$的简单性与$ \ Mathcalb_π$上的动作的（拓扑）freeness之间获得了类似的关系。一路上，我们扩展了connes-sullivan，并为单一表示$π$的平均属性提供了平均属性，并将其与$ c^*_π（γ）$的简单性和独特的跟踪属性相关联。
A discrete group $Γ$ is C*-simple if the C*-algebra $C_λ^*(Γ)$ generated by the range of the left regular representation $λ$ on $\ell^2(Γ)$ is simple. In this case, $Γ$ acts faithfully on the Furstenberg boundary $\partial_FΓ$ and there is a unique trace on $C_λ^*(Γ)$. In this paper we study the unique trace property for the C*-algebra $C_π^*(Γ)$ generated by the range of an arbitrary unitary representation $π: Γ\to B(H_π) $ and relate it to the faithfulness of the action of $Γ$ on the Furstenberg-Hamana boundary $\mathcal B_π$. Similar relation is obtained between simplicity of $C_π^*(Γ)$ and (topological) freeness of the action of $Γ$ on $\mathcal B_π$. Along the way, we extend the Connes-Sullivan and Powers averaging properties for a unitary representation $π$ and relate them to simplicity and unique trace property of $C^*_π(Γ)$.
