Journal of Operator Theory

Volume 46, Issue 2, Fall 2001 pp. 411-433.

Skew products and crossed products by coactions

Authors: S. Kaliszewski (1), John Quigg (2), and Iain Raeburn (3)
Author institution: (1) Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA
(2) Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA
(3) Department of Mathematics, University of Newcastle, NSW 2308, Australia

Summary: Given a labeling $c$ of the edges of a directed graph $E$ by elements of a discrete group $G$, one can form a skew-product graph $E\cross_c G$. We show, using the universal properties of the various constructions involved, that there is a coaction $\delta$ of $G$ on $C^*(E)$ such that $C^*(E\cross_c G)$ is isomorphic to the crossed product $C^*(E)\cross_\delta G$. This isomorphism is equivariant for the dual action $\what\delta$ and a natural action $\gamma$ of $G$ on $C^*(E\times_c G)$; following results of Kumjian and Pask, we show that $$ C^*(E\times_c G)\times_\gamma G \cong C^*(E\times_c G)\times_{\gamma,r}G \cong C^*(E)\otimes\K(\ell^2(G)),$$ and it turns out that the action $\gamma$ is always amenable. We also obtain corresponding results for $r$-discrete groupoids $Q$ and continuous homomorphisms $c\colon Q\to G$, provided $Q$ is amenable. Some of these hold under a more general technical condition which obtains whenever $Q$ is amenable or second-countable.

Keywords: $C^*$-algebra, coaction, skew product, directed graph, groupoid, duality

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