PreprintProductsystem models for twisted \(C^*\!\)algebras of topological higherrank graphsBecky Armstrong and Nathan BrownloweAbstractWe use product systems of \(C^*\!\)correspondences to introduce twisted \(C^*\!\)algebras of topological higherrank graphs. We define the notion of a continuous \(\mathbb{T}\)valued \(2\)cocycle on a topological higherrank graph, and present examples of such cocycles on large classes of topological higherrank graphs. To every proper, sourcefree topological higherrank 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 higherrank graph, Cuntz–Pimsner algebra.AMS Subject Classification: Primary 46L05.
This paper is available as a pdf (596kB) file. It is also on the arXiv: arxiv.org/abs/1706.09358.
