Higher-rank graph algebras are iterated 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 $\mathcal{T}{C}^{\ast }(\Lambda )$, may be realised as the Toeplitz algebra of a Hilbert $\mathcal{T}{C}^{\ast }({\Lambda }^i)$-bimodule. When $\Lambda$ is locally-convex, we show that the Cuntz-Krieger algebra of $\Lambda$, which we denote by $C^{\ast }(\Lambda )$, may be realised as the Cuntz-Pimsner algebra of a Hilbert $C^{\ast }({\Lambda }^i)$-bimodule. Consequently, $\mathcal{T}{C}^{\ast }(\Lambda )$ and $C^{\ast }(\Lambda )$ may be viewed as iterated Toeplitz and iterated Cuntz-Pimsner algebras over $c_0({\Lambda }^0)$ respectively.

arXiv comment38 pages, corrected an error in the proof of Lemma 4.13, added Section 5 (Relationships to other constructions), added Appendix A