We study strong compactly aligned product systems of ℤ+N over a C*-algebra A. We provide a description of their Cuntz-Nica-Pimsner algebra in terms of tractable relations coming from ideals of A. This approach encompasses product systems where the left action is given by compacts, as well as a wide class of higher rank graphs (beyond row-finite). Moreover we analyze higher rank factorial languages and their C*-algebras. Many of the rank one results in the literature find here their higher rank analogues. In particular, we show that the Cuntz-Nica-Pimsner algebra of a higher rank sofic language coincides with the Cuntz-Krieger algebra of its unlabeled follower set higher rank graph. However, there are also differences. For example, the Cuntz-Nica-Pimsner can lie in-between the first quantization and its quotient by the compactly supported operators.
Bibliographical noteFunding Information:
The first author was partially supported by an Azrieli international postdoctoral fellowship, and an Ontario trillium scholarship. Work on this project has been undertaken during a visit of the first author at Newcastle University, funded by the School of Mathematics and Statistics.
The first author is grateful to Ian Putnam for several discussions on sofic shifts and follower set graphs. The first author is also grateful for the support and hospitality of the Mathematics Department of University of Victoria, for a visit during which work on this project was conducted. The second author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme ?Operator algebras: subfactors and their applications? where work on this paper was undertaken. The authors would like to thank Valentin Deaconu for corrections provided
This work was supported by EPSRC grant no EP/K032208/1. Acknowledgements
© 2021, The Hebrew University of Jerusalem.
ASJC Scopus subject areas
- Mathematics (all)