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 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 Λ, and continuous T-valued 2-cocycle c on Λ, we associate a product system X of C0(Λ0)-correspondences built from finite paths in Λ. We define the twisted Cuntz–Krieger algebra C⁎(Λ,c) to be the Cuntz–Pimsner algebra O(X), and we define the twisted Toeplitz algebra TC⁎(Λ,c) to be the Nica–Toeplitz algebra NT(X). We also associate to Λ and c a product system Y of C0(Λ∞)-correspondences built from infinite paths. We prove that there is an embedding of TC⁎(Λ,c) into NT(Y), and an isomorphism between C⁎(Λ,c) and O(Y).