For a finitely generated group [Formula: see text] acting on a metric space [Formula: see text], Roe defined the warped space [Formula: see text], which one can view as a kind of large scale quotient of [Formula: see text] by the action of [Formula: see text]. In this paper, we generalize this notion to the setting of actions of arbitrary groups on large scale spaces. We then restrict our attention to what we call coarsely discontinuous actions by coarse equivalences and show that for such actions the group [Formula: see text] can be recovered as an appropriately defined automorphism group [Formula: see text] when [Formula: see text] satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group [Formula: see text] on a discrete bounded geometry metric space [Formula: see text] there is a relation between the maximal Roe algebras of [Formula: see text] and [Formula: see text], namely that there is a ∗-isomorphism [Formula: see text], where [Formula: see text] is the ideal of compact operators. If [Formula: see text] has Property A and [Formula: see text] is amenable, then [Formula: see text] has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

中文翻译：

---

群行动的粗商和最大 Roe 代数

对于作用于度量空间[公式：参见文本]的有限生成群[公式：参见文本]，Roe定义了扭曲空间[公式：参见文本]，可以将其视为[公式：参见文本]的一种大规模商。见文本] 通过 [公式：见文本] 的作用。在本文中，我们将这一概念推广到在大尺度空间上设置任意组的动作。然后，我们通过粗等价将注意力限制在我们所谓的粗略不连续动作上，并表明对于这样的动作，当 [公式：见文本] 时，可以将组 [公式：见文本] 恢复为适当定义的自同构群 [公式：见文本] ] 满足大规模连通性条件。我们表明，对于离散有界几何度量空间 [公式：见文] [公式：见文] 和 [公式：见文] 的最大 Roe 代数之间存在关系，即存在 ∗-同构 [公式：见文]，其中 [公式：见文] 为紧凑型运营商的理想之选。如果 [Formula: see text] 具有属性 A 并且 [Formula: see text] 是可服从的，则 [Formula: see text] 具有属性 A，因此最大 Roe 代数和全叉积可以替换为通常的 Roe 代数和分别在上述等式中减少交叉积。