Product-system models for twisted $C^*$-algebras of topological higher-rank graphs

Abstract

We use product systems of $C^*$-correspondences to introduce twisted $C^*$-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 $C^*(\Lambda,c)$ to be the Cuntz--Pimsner algebra $\mathcal{O}(X)$, and we define the twisted Toeplitz algebra $\mathcal{T} C^*(\Lambda,c)$ to be the Nica--Toeplitz algebra $\mathcal{NT}(X)$. 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 $\mathcal{T} C^*(\Lambda,c)$ into $\mathcal{NT}(Y)$, and an isomorphism between $C^*(\Lambda,c)$ and $\mathcal{O}(Y)$.