Doctor of Philosophy
School of Mathematics and Applied Statistics
Maloney, Ben, Semigroup actions on higher-rank graphs and their graph C*- algebras, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2012. http://ro.uow.edu.au/theses/3792
Higher-rank graphs and their C∗-algebras were introduced by Kumjian and Pask as a graphical means to provide combinatorial models of the Cuntz-Krieger families of Robertson and Steger. If a group G acts on a directed graph E, the universal property of C∗(E) shows that this action on the graph induces an action of G on C∗(E), and hence the crossed product C∗(E) ⋊ G can be formed. A theorem of Kumjian and Pask says that if a group G acts freely on a directed graph E, then the associated crossed product C∗(E) ⋊ G of the graph algebra is stably isomorphic to the graph algebra C∗(G\E) of the quotient graph. A similar result was established by Pask, Raeburn and Yeend for certain actions of semigroups on directed graphs.
The main purpose of this thesis is to prove this relationship in the most general case: that is, for an Ore semigroup S and a free action α : S → End Σ on a higher-rank graph Σ that admits a fundamental domain, the crossed product C∗(Σ) X α* S is stably isomorphic to C∗(S\Σ). We will show that there exists an isomorphism from C∗(Σ) X α* S onto C∗(S\Σ)⊗ K(ɭ2(S)). That is, the associated crossed product C∗(Σ) X α* S of the graph algebra is stably isomorphic to the graph algebra C∗(S\Σ) of the quotient k-graph.
The isomorphism is realised in three stages: the method that we use is an im- provement of the method used by Pask, Raeburn and Yeend using the full weight of a dilation result by Laca. The extra generality means that new proofs at each stage were required. We first prove a version of the Gross-Tucker theorem for semi- group actions on higher-rank graphs. Second, we give a new formulation of Laca's result involving endomorphisms of Ore semigroups actions on C∗-algebras that is specially tailored to the higher-rank graph case. By doing so, we also have man- aged to shorten the proof for the directed graph case. We also generalise the work of Kaliszewski, Quigg and Raeburn to recognise Cuntz-Krieger families of crossed products in associated tensor products, and provide an explicit isomorphism for the relationship. For each step we have managed to give an explicit isomorphism by breaking down the steps used in the argument of Pask, Raeburn and Yeend. We apply multiplier algebra techniques to produce an isomorphism between the semi- group crossed product and the dilated group crossed product. In order to do this, we provide new results on convergence in the multiplier algebra of a higher-rank graph C∗-algebra.
Finally, we give some applications of our results, such as criteria for simplicity and pure infiniteness of skew-product graph C∗-algebras. We consider a dual k- graph construction that allows us to consider non-free actions on a k-graph that are free on some associated dual graph. We extend the idea of primitivity, and give some equivalent conditions to aperiodicity and cofinality that are more straightforward to check. We also give simplicity criteria for the fixed-point algebra of the gauge action. As an alternative to our efficient use of Laca's dilation theory for endomorphic actions of Ore semigroups on C∗-algebras we could have performed the complicated direct limit construction. We feel these calculations are of independent interest, and so they are also provided.