Crossed products and twisted $k$-graph algebras

TL;DR

This paper explores how automorphisms of $k$-graphs induce crossed product $C^*$-algebras and examines their interaction with twisted $k$-graph algebras using cohomology, providing new insights and examples.

Contribution

It demonstrates the interaction between crossed products and twisted $k$-graph $C^*$-algebras via cohomology and long exact sequences.

Findings

Crossed products correspond to $(k+1)$-graph algebras.

The interaction with twists is characterized by cohomology sequences.

Examples illustrate the theoretical results.

Abstract

An automorphism $\beta$ of a $k$-graph $\Lambda$ induces a crossed product $C^* ( \Lambda ) \rtimes_\beta \mathbb{Z}$ which is isomorphic to a $(k+1)$-graph algebra $C^* ( \Lambda \times_\beta \mathbb{Z})$. In this paper we show how this process interacts with $k$-graph $C^*$-algebras which have been twisted by an element of their second cohomology group. This analysis is done using a long exact sequence in cohomology associated to this data. We conclude with some examples