论文标题
组$ c^*$ - 作用在树上的本地紧凑型组的代数
Group $C^*$-algebras of locally compact groups acting on trees
论文作者
论文摘要
我们研究组$ c^*$ - 代数$ c^*_ {l^{p+}}（g）$ - 由$ l^p $ - 统一表示的矩阵系数的矩阵系数的构建 - 属于本地紧凑的组的$ g $ g $在（半型）同质树上的均等小型树。这些组$ c^*$ - 代数位于通用和简化组$ c^*$ - 代数之间。通过直接调查这些$ l^p $ - 整合性属性，我们首先表明，对于每一个非紧密的，封闭的亚组$ g $，$ \ mathrm {aut}（aut}（t）$的（半）同质的树$ t $都可以在边界$ \ \ paus $ 2 $ 2 \ le q le q le q p p p p \ p p p p p \ p p p p p \ p p p p \ p p p。 $ c^*_ {l^{p+}}}（g）\ tweatheadRightArrow c^*_ {l^{q+}}}（g）$不是注入的。这谴责了Samei和Wiersma的结果。我们证明，在其他假设的情况下，$ g $在$ t $上进行了传递作用，并且它具有山雀的独立性，$ c^*$ - 代数$ c^*_ {l^{l^{p+}}（g）$是唯一来自$ g $ -g $ -g $ -invariant Idealss $ gene gene gene gere-s-al的组$ c^*$ - algebras。此外，我们表明，像以前一样给定一个组$ g $，每组$ c^*$ - 代数$ c^*_μ（g）$可区分的（作为$ c^*$ - 代数）与通用$ c^*$ - $ g $的$ g $的代数 - $ {}^*$ - 简化组$ c^*$ - $ g $的代数。
We study the group $C^*$-algebras $C^*_{L^{p+}}(G)$ - constructed from $L^p$-integrability properties of matrix coefficients of unitary representations - of locally compact groups $G$ acting on (semi-)homogeneous trees of sufficiently large degree. These group $C^*$-algebras lie between the universal and the reduced group $C^*$-algebra. By directly investigating these $L^p$-integrability properties, we first show that for every non-compact, closed subgroup $G$ of the automorphism group $\mathrm{Aut}(T)$ of a (semi-)homogeneous tree $T$ that acts transitively on the boundary $\partial T$ and every $2 \leq q < p \leq \infty$, the canonical quotient map $C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ is not injective. This reproves a result of Samei and Wiersma. We prove that under the additional assumptions that $G$ acts transitively on $T$ and that it has Tits' independence property, the group $C^*$-algebras $C^*_{L^{p+}}(G)$ are the only group $C^*$-algebras coming from $G$-invariant ideals in the Fourier-Stieltjes algebra $B(G)$. Additionally, we show that given a group $G$ as before, every group $C^*$-algebra $C^*_μ(G)$ that is distinguishable (as a group $C^*$-algebra) from the universal group $C^*$-algebra of $G$ and whose dual space $C^*_μ(G)^*$ is a $G$-invariant ideal in $B(G)$ is abstractly ${}^*$-isomorphic to the reduced group $C^*$-algebra of $G$.
