Journal of Operator Theory

Volume 74, Issue 1, Summer 2015 pp. 213-245.

Nuclearity and exactness for groupoid crossed products

Authors: Scott M. LaLonde
Author institution:Department of Mathematics, The University of Texas at Tyler, 3900 University Boulevard, Tyler, TX 75799, U.S.A.

Summary: Let $(\mathcal{A}, G, \alpha)$ be a groupoid dynamical system. We show that if $G$ is assumed to be measurewise amenable and the section algebra $A = \Gamma_0(G^{\scriptscriptstyle{(0)}}, \mathcal{A})$ is nuclear, then the associated groupoid crossed product is also nuclear. This generalizes an earlier result of Green for crossed products by locally compact groups. We also extend a related result of Kirchberg to groupoids. In particular, if $A$ is exact and $G$ is amenable, then we show that $\mathcal{A} \rtimes G$ is exact.

DOI: http://dx.doi.org/10.7900/jot.2014jun06.2032
Keywords: groupoid crossed product, $C_0(X)$ -algebra, nuclearity, exactness, exact groupoid

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