Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 349-371.

A Steinberg algebra approach to etale groupoid $C^*$-algebras

Authors: Lisa Orloff Clark (1), Joel Zimmerman (2)
Author institution: (1) School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
(2) School of Mathematics and Applied Statistics, University of Wollongong, Northfields Avenue, Wollongong 2522, Australia

Summary: We construct the full and reduced $C^*$-algebras of an ample groupoid from its complex Steinberg algebra. We also show that our construction gives the same $C^*$-algebras as the standard constructions. In the last section, we consider an arbitrary locally compact, second-countable, etale groupoid, possibly non-Hausdorff. Using the techniques developed for Steinberg algebras, we show that every $*$-homomorphism from Connes' space of functions to $B(\mathcal{H})$ is automatically $I$-norm bounded. Previously, this was only known for Hausdorff groupoids.

DOI: http://dx.doi.org/10.7900/jot.2022mar31.2446
Keywords: groupoid $C^*$-algebra, Steinberg algebra

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