Stable rank and real rank of compact transformation group $C^\ast$-algebras
Volume 175 / 2006
Studia Mathematica 175 (2006), 103-120
MSC: 22D15, 46L35, 54H15.
DOI: 10.4064/sm175-2-1
Abstract
Let $(G,X)$ be a transformation group, where $X$ is a locally compact Hausdorff space and $G$ is a compact group. We investigate the stable rank and the real rank of the transformation group $C^\ast$-algebra $C_0(X)\rtimes G$. Explicit formulae are given in the case where $X$ and $G$ are second countable and $X$ is locally of finite $G$-orbit type. As a consequence, we calculate the ranks of the group $C^\ast$-algebra $C^\ast(\mathbb{R}^n \rtimes G)$, where $G$ is a connected closed subgroup of $\mbox{SO}(n)$ acting on $\mathbb{R}^n$ by rotation.
