(Non)exotic completions of the group algebras of isotropy groups

TL;DR

This paper investigates the nature of the reduced norm on group algebras of isotropy groups within étale groupoids, revealing conditions under which it aligns with or diverges from the reduced C*-norm, with implications for KMS states.

Contribution

It demonstrates that the reduced norm induces a C*-norm on isotropy group algebras, identifying cases where this norm is exotic or coincides with the reduced norm, especially for certain classes of graded groupoids.

Findings

The reduced norm induces a C*-norm on isotropy group algebras.

The norm coincides with the reduced norm for transformation groupoids.

The norm can be exotic for groupoids of germs, but matches the reduced norm for specific classes like partial actions and semidirect products.

Abstract

Motivated by the problem of characterizing KMS states on the reduced C$^*$-algebras of \'etale 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.