Higher-rank graph algebras are iterated Cuntz-Pimsner algebras

TL;DR

This paper demonstrates that higher-rank graph algebras can be constructed as iterated Cuntz-Pimsner algebras, providing a new perspective on their structure through bimodule frameworks.

Contribution

It introduces a novel realization of higher-rank graph algebras as iterated Cuntz-Pimsner algebras, extending the understanding of their algebraic structure.

Findings

Toeplitz-Cuntz-Krieger algebra as Toeplitz algebra of a bimodule

Cuntz-Krieger algebra as Cuntz-Pimsner algebra of a bimodule

Algebras viewed as iterated Toeplitz and Cuntz-Pimsner algebras

Abstract

Given a finitely aligned $k$-graph $\Lambda$, we let $\Lambda^i$ denote the $(k-1)$-graph formed by removing all edges of degree $e_i$ from $\Lambda$. We show that the Toeplitz-Cuntz-Krieger algebra of $\Lambda$, denoted by $\mathcal{T}C^*(\Lambda)$, may be realised as the Toeplitz algebra of a Hilbert $\mathcal{T}C^*(\Lambda^i)$-bimodule. When $\Lambda$ is locally-convex, we show that the Cuntz-Krieger algebra of $\Lambda$, which we denote by $C^*(\Lambda)$, may be realised as the Cuntz-Pimsner algebra of a Hilbert $C^*(\Lambda^i)$-bimodule. Consequently, $\mathcal{T}C^*(\Lambda)$ and $C^*(\Lambda)$ may be viewed as iterated Toeplitz and iterated Cuntz-Pimsner algebras over $c_0(\Lambda^0)$ respectively.