Higher-rank graph algebras are iterated Cuntz-Pimsner algebras

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