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

TL;DR

This paper develops a framework for analyzing higher rank operator algebras associated with factorial languages, extending known rank-one results and providing new descriptions of their C*-algebras using ideals and graph structures.

Contribution

It introduces a description of Cuntz-Nica-Pimsner algebras for higher rank product systems and extends rank-one results to higher rank factorial languages and graphs.

Findings

Cuntz-Nica-Pimsner algebra coincides with higher rank graph C*-algebra for sofic languages

Provides tractable relations from ideals of the base algebra

Shows differences between Cuntz-Nica-Pimsner and quotient by compact operators

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.