arXiv subjects
Publications and source records attributed to Ben Maloney.
We consider conditions on a $k$-graph $Λ$, a semigroup $S$ and a functor $η: Λ\to S$ which ensure that the $C^*$-algebra of the skew-product graph $Λ\times_ηS$ is simple. Our results allow give some necessary and sufficient conditions for the AF-core of a $k$-graph $C^{*}$-algebra to be simple.
We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the $C^*$-algebra of the graph. We show that the crossed product by this action is stably isomorphic to the $C^*$-algebra of a quotient graph. Our main tool is Laca's dilation theory for endomorphic actions of Ore semigroups on $C^*$-algebras, which embeds such an action in an automorphic action of the enveloping group on a larger $C^*$-algebra.
We introduce the concept of a 1-coaligned $k$-graph and prove that the shift maps of a $k$-graph pairwise *-commute if and only if the $k$-graph is 1-coaligned. We then prove that for 2-graphs $Λ$ generated from basic data *-commuting shift maps is equivalent to a condition that implies that $C^*(Λ)$ is simple and purely infinite. We then consider full shift spaces and introduce a condition on a block map which ensures the associated sliding block code *-commutes with the shift.