Higher-rank graph algebras are iterated Cuntz-Pimsner algebras

DOI10.17879/85169761673zbMATH Open1432.46038arXiv1711.01698MaRDI QIDQ5224443

James Fletcher

Publication date: 23 July 2019

Abstract: Given a finitely aligned $k$-graph $Lambda$, we let $Lambd{a}^i$ 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 $mathcalT{C}^{*}(Lambda)$, may be realised as the Toeplitz algebra of a Hilbert $mathcalT{C}^{*}(Lambd{a}^i)$-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 $C^{*}(Lambd{a}^i)$-bimodule. Consequently, $mathcalT{C}^{*}(Lambda)$ and $C^{*}(Lambda)$ may be viewed as iterated Toeplitz and iterated Cuntz-Pimsner algebras over $c_0(Lambd{a}^0)$ respectively.

Full work available at URL: https://arxiv.org/abs/1711.01698

zbMATH Keywords

Cuntz-Pimsner algebrafinitely aligned $k$-graphToeplitz-Cuntz-Krieger algebra

Mathematics Subject Classification ID

General theory of (C^*)-algebras (46L05) Graph theory (05C99)

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