极小理想与约尔当群的约简群C*-代数
Maximal ideals of reduced group C*-algebras and Thompson's groups
AI总结：
本文研究了约简群C*-代数的极小理想与约尔当群的极小理想之间的关系，证明了约尔当群T的约简C*-代数有唯一的极小理想，若F群可 amenable，则T的约简C*-代数有无限多个理想。
摘要：
给定一个从C*-代数B到其C*-子代数A的条件期望P，我们观察到通过P诱导理想以及我们称之为co-induction的映射，构成了A和B的理想格之间的Galois连接。利用该Galois连接的性质，我们证明，给定一个离散群G及其在Furstenberg边界上的稳定子群G_x，诱导给出C*(G_x)的极小co-induced理想与C*_r(G)的极小理想之间的双射。作为应用，我们证明了约尔当群T的约简C*-代数有唯一的极小理想。进一步地，我们证明，如果约尔当群F是amenable的，则T的约简C*-代数有无限多个理想。
英文摘要：
Given a conditional expectation $P$ from a C*-algebra $B$ onto a C*-subalgebra $A$, we observe that induction of ideals via $P$, together with a map which we call co-induction, forms a Galois connection between the lattices of ideals of $A$ and $B$. Using properties of this Galois connection, we show that, given a discrete group $G$ and a stabilizer subgroup $G_x$ for the action of $G$ on its Furstenberg boundary, induction gives a bijection between the set of maximal co-induced ideals of $C^*(G_x)$ and the set of maximal ideals of $C^*_r(G)$. As an application, we prove that the reduced C*-algebra of Thompson's group $T$ has a unique maximal ideal. Furthermore, we show that, if Thompson's group $F$ is amenable, then $C^*_r(T)$ has infinitely many ideals.
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