The path space of a higher-rank graph

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

30 pages, all figures drawn with TikZ/PGF. Updated numbering and minor corrections to coincide with published version. Updated 29-Feb-2012 to fix a compiling error which resulted in the arXiv PDF output containing two copies of the article

Cited by in corpus (2)

• On some fundamental results about higher-rank graphs and their C*-algebras
• Primitive ideal space of Higher-rank graph $C^{\ast }$-algebras and decomposability