The path space of a higher-rank graph

Samuel B. G. Webster ¹

¹ School of Mathematics and Applied Statistics University of Wollongong Wollongong, NSW 2522, Australia

Studia Mathematica, Tome 204 (2011) no. 2, pp. 155-185

Résumé

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

Détail

DOI : 10.4064/sm204-2-4

Keywords: construct locally compact hausdorff topology path space finitely aligned k graph mit lambda identify boundary path space partial mit lambda spectrum commutative * subalgebra mit lambda * mit lambda using construction similar farthing construct finitely aligned k graph widetilde mit lambda sources which mit lambda embedded partial mit lambda homeomorphic subset partial widetilde mit lambda mit lambda row finite identify * mit lambda full corner * widetilde mit lambda deduce mit lambda isomorphic corner widetilde mit lambda lastly isomorphism implements homeomorphism between boundary path spaces