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

Becky Armstrong and Nathan Brownlowe

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 $\mathrm{\Lambda }$, and continuous $\mathbb{T}$-valued $2$-cocycle $c$ on $\mathrm{\Lambda }$, we associate a product system $X$ of $C_0({\mathrm{\Lambda }}^0)$-correspondences built from finite paths in $\mathrm{\Lambda }$. We define the twisted Cuntz–Krieger algebra $C^{\ast }(\mathrm{\Lambda },c)$ to be the Cuntz–Pimsner algebra $\mathcal{O}(X)$, and we define the twisted Toeplitz algebra $\mathcal{T}{C}^{\ast }(\mathrm{\Lambda },c)$ to be the Nica–Toeplitz algebra $\mathcal{NT}(X)$. We also associate to $\mathrm{\Lambda }$ and $c$ a product system $Y$ of $C_0({\mathrm{\Lambda }}^{\mathrm{\infty }})$-correspondences built from infinite paths. We prove that there is an embedding of $\mathcal{T}{C}^{\ast }(\mathrm{\Lambda },c)$ into $\mathcal{NT}(Y)$, and an isomorphism between $C^{\ast }(\mathrm{\Lambda },c)$ and $\mathcal{O}(Y)$.

Keywords: C*-algebra, product system, topological higher-rank graph, Cuntz–Pimsner algebra.

AMS Subject Classification: Primary 46L05.