Strength of convergence in the orbit space of a groupoid (Q549825)

This paper considers second-countable locally-compact Hausdorff groupoids with Haar systems, \(G\), and proves that a sequence \(\{x_n\}\) in the unit space \(G_0\) of \(G\) converging \(k\)-times in the orbit space to \(z\in G_0\) is equivalent to: (i) a measure-theoretic accumulation along the orbits; and (ii) the lower multiplicity number associated to a sequence of induced representations of the groupoid \(C^*\)-algebra of \(G\) being \(k\). This is a generalization of a result from [\textit{R. J. Archbold} and \textit{A. an Huef}, J. Funct. Anal. 235, No. 1, 90--121 (2006; Zbl 1092.22005)] to principal groupoids, and leads to a new class of examples exhibiting \(k\)-times convergence in groupoids not based on transformation groups. The specific examples discussed are of path groupoids [\textit{A. Kumjian, D. Pask, I. Raeburn} and \textit{J. Renault}, J. Funct. Anal. 144, No. 2, 505--541 (1997; Zbl 0929.46055)] and can easily be used to find groupoids whose associated \(C^*\)-algebra has a non-Hausdorff spectrum and has irreducible representations that have distinct upper and lower multiplicity counts.