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arXiv · 1803.11260

Operator algebras for higher rank analysis and their application to factorial languages

Adam Dor-On·Evgenios T. A. Kakariadis
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Abstract

We study strong compactly aligned product systems of $\mathbb{Z}_+^N$ 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.

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Adam Dor-On, Evgenios T. A. Kakariadis. 2018-03-29. Operator algebras for higher rank analysis and their application to factorial languages. https://arxiv.org/abs/1803.11260

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