Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 55-91.

Exotic $C^*$-completions of etale groupoids

Authors: Mathias Palmstrom
Author institution: Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, 7491, Norway

Summary: We generalize the ideal completions of countable discrete groups, as introduced by Brown and Guentner, to second-countable Hausdorff etale groupoids. Specifically, to every pair consisting of an algebraic ideal in the algebra of bounded Borel functions on the groupoid and a non-empty family of quasi-invariant measures on the unit space, we construct a $C^*$-algebra in a way which naturally encapsulates the constructions of the full and reduced groupoid $C^*$-algebras. We investigate the connection between these constructions and the Haagerup property, and use the construction to show the existence of many exotic groupoid $C^*$-algebras for certain classes of groupoids.

DOI: http://dx.doi.org/10.7900/jot.2024nov28.2490
Keywords: groupoid $C^*$-algebras, unitary representation of groupoids

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