Product-system models for twisted $C^{\ast }$-algebras of topological higher-rank graphs

Abstract

We use product systems of $C^{\ast }$-correspondences to introduce twisted $C^{\ast }$-algebras of topological higher-rank graphs. We define the notion of a continuous $\mathbb{T}$-valued $2$-cocycle on a topological higher-rank graph, and present examples of such cocycles on large classes of topological higher-rank graphs. To every proper, source-free topological higher-rank graph $\Lambda$, and continuous $\mathbb{T}$-valued $2$-cocycle $c$ on $\Lambda$, we associate a product system $X$ of $C_0({\Lambda }^0)$-correspondences built from finite paths in $\Lambda$. We define the twisted Cuntz--Krieger algebra $C^{\ast }(\Lambda ,c)$ to be the Cuntz--Pimsner algebra $\mathcal{O}(X)$, and we define the twisted Toeplitz algebra $\mathcal{T}{C}^{\ast }(\Lambda ,c)$ to be the Nica--Toeplitz algebra $\mathcal{N}\mathcal{T}(X)$. We also associate to $\Lambda$ and $c$ a product system $Y$ of $C_0({\Lambda }^{\infty })$-correspondences built from infinite paths. We prove that there is an embedding of $\mathcal{T}{C}^{\ast }(\Lambda ,c)$ into $\mathcal{N}\mathcal{T}(Y)$, and an isomorphism between $C^{\ast }(\Lambda ,c)$ and $\mathcal{O}(Y)$.

30 pages. This version matches the version in the Journal of Mathematical Analysis and Applications

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• C*-algebras associated to product systems of Hilbert bimodules
• Groupoid models for the C*-algebras of topological higher-rank graphs
• Crossed products of k-graph C*-algebras by Z^l
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Cited by in corpus (3)

• Twisted Steinberg algebras
• Iterating the Cuntz-Nica-Pimsner construction for compactly aligned product systems
• Simplicity of twisted C*-algebras of Deaconu--Renault groupoids