AI 中文总结
本文基于Tits的W-距离定义引入C*-代数在Coxeter系统上的分解概念，构造相关Fock希尔伯特模与约化C*-代数等，证明协变代数的泛性质，得到部分新的逼近性质。
AI 中文摘要
我们基于Tits的W-距离定义，引入并研究C*-代数在Coxeter系统上的分解概念。当此类分解带有合适的条件期望时，我们构造了相关的Fock希尔伯特模与约化C*-代数A^r，这统一了Voiculescu[Voi85]和Caspers-Fima[CF17]的构造，并提供了离散群G作用于建筑这一情形的非交换类比，其中A^r ≅ C*_r(G)。我们构造了协变C*-代数𝒞(i)⊃A^r，作为交叉积C(Ω)⋊_r G ⊃ C*_r(G)的非交换类比，这里Ω是Caprace和Lécureux[CL11]给出的局部有限建筑的极小组合紧化。遵循Hasegawa[Has17]和Klisse[Kli25]的方法，我们证明了协变代数的泛性质。这一结构结果给出了不同的逼近性质，其中部分性质即使在群情形下也是新的。
英文摘要
We introduce and study a notion of decomposition of a C$^*$-algebra over a Coxeter system based on Tits' definition of a $W$-distance. When such a decomposition comes with suitable conditional expectations, we build an associated Fock Hilbert module and reduced C$^*$-algebra $A^r$. This unifies constructions of Voiculescu [Voi85] and Caspers--Fima [CF17] and provides a noncommutative analogue of the situation of a discrete group $G$ acting on a building, in which case $A^r \cong C^*_r(G)$. We construct covariance C$^*$-algebras $\mathscr{C}(i)\supset A^r$ as a noncommutative analogue of the crossed product $C(Ω)\rtimes_r G \supset C^*_r(G) $ where $Ω$ is Caprace and Lécureux's minimal combinatorial compactification [CL11] of the locally finite building. Following the approach of Hasegawa [Has17] and Klisse [Kli25], we prove a universal property for the covariance algebras. This structural result yields different approximation properties, some of which are new even for the group case.
