## 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.

: Primary 46L05.

This paper is available as a pdf (596kB) file. It is also on the arXiv: arxiv.org/abs/1706.09358.

 Friday, July 7, 2017