(Non)exotic completions of the group algebras of isotropy groups
Johannes Christensen, Sergey Neshveyev
TL;DR
The paper studies when KMS states on reduced groupoid C*-algebras $C^*_r(\mathcal{G})$ factor through the full algebra $C^*({\mathcal{G}})$ by focusing on isotropy algebras. It introduces an exotic C*-norm $\|\cdot\|_{e}$ on $\mathbb{C}{\mathcal{G}}^{x}_{x}$, characterizes factorization through $C^*_r(\mathcal{G})$ via representations whose Induced representations factor through the reduced algebra, and shows $\|h\|_{e}$ can be described as $\|h\|_{e}=\inf\{\|f\|_{r}:\eta_x(f)=h\}$ with a limit form involving compressions by $q_V$. The main focus is when $\|\cdot\|_{e}$ coincides with the reduced norm $\|\cdot\|_{r}$, which holds for transformation groupoids and often for graded groupoids under suitable hypotheses, including exactness and amenable isotropy in semidirect or partial-action constructions. This yields precise criteria for when states and weights with diagonal centralizers factor through $C^*_r(\mathcal{G})$, and demonstrates Morita invariance of these norm-equality properties across a broad class of graded and non-Hausdorff groupoids.
Abstract
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.