Norm upper-semicontinuity of functions supported on open abelian isotropy in \'etale groupoids (a corrigendum to "Reconstruction of groupoids and C*-rigidity of dynamical systems," Adv. Math 390 (2021), 107923)

TL;DR

This paper proves that in certain étale groupoids with abelian isotropy, the norms of elements supported on the isotropy's interior vary upper semicontinuously, correcting a previous error in the literature.

Contribution

It establishes the upper semicontinuity of norms for elements supported on the isotropy's interior in étale groupoids, rectifying prior inaccuracies.

Findings

Norms of elements supported on the isotropy's interior are upper semicontinuous.

The result corrects a previous error in the literature.

Provides a rigorous foundation for analyzing groupoid C*-algebras.

Abstract

We consider \'etale Hausdorff groupoids in which the interior of the isotropy is abelian. We prove that the norms of the images under regular representations, of elements of the reduced groupoid $C^*$-algebra whose supports are contained in the interior of the isotropy vary upper semicontinuously. This corrects an error in [T.M. Carlsen, E. Ruiz, A. Sims and M. Tomforde, "Reconstruction of groupoids and C*-rigidity of dynamical systems," Adv. Math 390 (2021), 107923].