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)

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].