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Separable representations, KMS states, and wavelets for higher-rank graphs
| Content Provider | Semantic Scholar |
|---|---|
| Author | Farsi, Carla Gillaspy, Elizabeth A. Kang, Sooran Packer, Judith A. |
| Copyright Year | 2015 |
| Abstract | Let $\Lambda$ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of $C^*(\Lambda)$ on certain separable Hilbert spaces of the form $L^2(X,\mu)$, by introducing the notion of a $\Lambda$-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, when $\Lambda$ is aperiodic, we obtain a faithful representation of $C^*(\Lambda)$ on $L^2(\Lambda^\infty, M)$, where $M$ is the Perron-Frobenius probability measure on the infinite path space $\Lambda^\infty$ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a $\Lambda$-semibranching function system gives rise to KMS states for $C^*(\Lambda)$. For the higher-rank graphs of Robertson and Steger, we also obtain a representation of $C^*(\Lambda)$ on $L^2(X, \mu)$, where $X$ is a fractal subspace of $[0,1]$ by embedding $\Lambda^{\infty}$ into $[0,1]$ as a fractal subset $X$ of $[0,1]$. In this latter case we additionally show that there exists a KMS state for $C^*(\Lambda)$ whose inverse temperature is equal to the Hausdorff dimension of $X$. Finally, we construct a wavelet system for $L^2(\Lambda^\infty, M)$ by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs. |
| Starting Page | 241 |
| Ending Page | 270 |
| Page Count | 30 |
| File Format | PDF HTM / HTML |
| DOI | 10.1016/j.jmaa.2015.09.003 |
| Alternate Webpage(s) | http://export.arxiv.org/pdf/1505.00485 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1505.00485v2.pdf |
| Alternate Webpage(s) | https://doi.org/10.1016/j.jmaa.2015.09.003 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
