# Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 347-382.

Groupoid algebras as covariance algebras

**Authors**:
Lisa Orloff Clark (1), James Fletcher (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 Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand

**Summary: **Suppose $\mathcal{G}$ is a second-countable locally compact Hausdorff \'{e}tale groupoid, $G$ is a discrete group containing a unital subsemigroup $P$, and $c:\mathcal{G}\rightarrow G$ is a continuous cocycle. We derive conditions on the cocycle such that the reduced groupoid $C^*$-algebra $C_\mathrm r^*(\mathcal{G})$ may be realised as the covariance algebra of a product system over $P$ with coefficient algebra $C_\mathrm r^*(c^{-1}(e))$. When $(G,P)$ is a quasi-lattice ordered group, we also derive conditions that allow $C_\mathrm r^*(\mathcal{G})$ to be realised as the Cuntz--Nica--Pimsner algebra of a compactly aligned product system.

**DOI: **http://dx.doi.org/10.7900/jot.2019aug22.2266

**Keywords: ** groupoid $C^*$-algebra, product system, covariance algebra

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