Motivated by the problem of characterizing KMS states on the reduced C$^*$-algebras of étale groupoids, we show that the reduced norm on these algebras induces a C$^*$-norm on the group algebras of the isotropy groups. This C$^*$-norm coincides with the reduced norm for the transformation groupoids, but, as follows from examples of Higson–Lafforgue–Skandalis, it can be exotic already for groupoids of germs associated with group actions. We show that the norm is still the reduced one for some classes of graded groupoids, in particular, for the groupoids associated with partial actions of groups and the semidirect products of exact groups and groupoids with amenable isotropy groups.

中文翻译：

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各向同性群的群代数的（非）奇异补全

受在 étale 群的约简 C$^*$-代数上表征 KMS 状态的问题的启发，我们表明这些代数上的约简范数在各向同性群的群代数上引发了一个 C$^*$-范数。这个 C$^*$-norm 与变换群的简化范数一致，但是，从 Higson-Lafforgue-Skandalis 的示例中可以看出，对于与群行为相关的细菌群群，它可能已经是奇异的了。我们表明，对于某些等级的类群，范数仍然是简化的范数，特别是对于与群的部分作用相关的群群和精确群的半直积和具有服从各向同性群的群群。