Purely infinite $C^{\ast }$-algebras associated to étale groupoids
Purely infinite $C^{\ast }$-algebras associated to étale groupoids
复制标题
与étale群胚相关的纯无限$C^{ast }$-代数
DOI：
10.1017/etds.2014.47
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发表时间：
2014
影响因子：
0.9
通讯作者：
Adam Sierakowski
中科院分区：
文献类型：
--
作者：
Jonathan S. Brown;L. O. Clark;Adam Sierakowski
Let $G$ be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if all the non-zero positive elements of $C_{0}(G^{(0)})$ are infinite in $C_{r}^{\ast }(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if every non-zero projection in $C_{0}(G^{(0)})$ is infinite in $C_{r}^{\ast }(G)$. We then show how this result applies to $k$-graph $C^{\ast }$-algebras. Finally, we investigate strongly purely infinite groupoid $C^{\ast }$-algebras.
Let $G$ be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if all the non-zero positive elements of $C_{0}(G^{(0)})$ are infinite in $C_{r}^{\ast }(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C_{r}^{\ast }(G)$ is purely infinite simple if and only if every non-zero projection in $C_{0}(G^{(0)})$ is infinite in $C_{r}^{\ast }(G)$. We then show how this result applies to $k$-graph $C^{\ast }$-algebras. Finally, we investigate strongly purely infinite groupoid $C^{\ast }$-algebras.