Strength of convergence in the orbit space of a groupoid

TL;DR

This paper explores the relationship between convergence notions in the orbit space of a groupoid and measure-theoretic accumulation, linking these to multiplicity numbers in groupoid C*-algebras.

Contribution

It establishes the equivalence between convergence strength and measure-theoretic accumulation in the orbit space of a groupoid, connecting these to representation multiplicities.

Findings

Convergence notions in the orbit space are equivalent to measure-theoretic accumulation.

Multiplicity numbers of induced representations are characterized by these convergence properties.

Abstract

Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {x_n} be a sequence in the unit space of G. We show that the notions of strength of convergence of {x_n} in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C*-algebra.