Product-system models for twisted \(C^*\!\)-algebras of topological higher-rank graphs

Becky Armstrong and Nathan Brownlowe

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 \(\mathcal{O}(Y)\).

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

AMS Subject Classification: Primary 46L05.