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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