The path space of a higher-rank graph

TL;DR

This paper constructs a topology on the path space of higher-rank graphs, identifies boundary-path spaces with spectra of certain subalgebras, and relates these structures through embeddings and isomorphisms, enhancing understanding of their $C^*$-algebraic properties.

Contribution

It introduces a new topology on the path space of finitely aligned $k$-graphs and establishes a homeomorphism between boundary-path spaces via graph embeddings and $C^*$-algebraic isomorphisms.

Findings

Constructed a locally compact Hausdorff topology on the path space.

Identified the boundary-path space as the spectrum of a subalgebra of the $C^*$-algebra.

Established a homeomorphism between boundary-path spaces of embedded graphs.

Abstract

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.