The path space of a higherrank graph
Abstract
We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$graph $\Lambda$. We identify the boundarypath 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 rowfinite, 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 boundarypath spaces.
 Publication:

arXiv eprints
 Pub Date:
 February 2011
 arXiv:
 arXiv:1102.1229
 Bibcode:
 2011arXiv1102.1229W
 Keywords:

 Mathematics  Operator Algebras;
 46L05
 EPrint:
 30 pages, all figures drawn with TikZ/PGF. Updated numbering and minor corrections to coincide with published version. Updated 29Feb2012 to fix a compiling error which resulted in the arXiv PDF output containing two copies of the article