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Journal of Operator Theory

Volume 68, Issue 1, Summer 2012  pp. 101-130.

Realising the $C^*$-algebra of a higher-rank graph as an Exel's crossed product

Authors Nathan Brownlowe
Author institution: School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522, Australia

Summary:  We use the boundary-path space of a finitely-aligned $k$-graph $\Lambda$ to construct a compactly-aligned product system $X$, and we show that the graph algebra $C^*(\Lambda)$ is isomorphic to the Cuntz-Nica-Pimsner algebra $\mathcal{NO}(X)$. In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be $\mathcal{NO}(X)$. We then compare this crossed product with other definitions in the literature.

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