The path space of a higher-rank graph

Details The path space of a higher-rank graph The path space of a higher-rank graph

2011-02-07

arxiv.org/abs/1102.1229v3

Studia Math. 204 (2011), 155-185

Mathematics

Operator Algebras

30 pages, all figures drawn with TikZ/PGF. Updated numbering and minor corrections to coincide with published version. Updated

Scientific paper

10.4064/sm204-2-4

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\Lambda$. We identify the boundary-path space $\partial\Lambda$ as the spectrum of a commutative $C^*$-subalgebra $D_\Lambda$ of $C^*(\Lambda)$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph $\wt\Lambda$ with no sources in which $\Lambda$ is embedded, and show that $\partial\Lambda$ is homeomorphic to a subset of $\partial\wt\Lambda$ . We show that when $\Lambda$ is row-finite, we can identify $C^*(\Lambda)$ with a full corner of $C^*(\wt\Lambda)$, and deduce that $D_\Lambda$ is isomorphic to a corner of $D_{\wt\Lambda}$. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.

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