The categorical properties of higher rank graphs and applications to their C*-algebras

The class of Cuntz-Krieger C*-algebras associated to higher rank graphs (k-graphs) represents an interesting field of study because of their similarities to the C*-algebras associated to directed graphs and also their dissimilarities which enable them to have much richer and complex structures. Here we develop some methods for determining Morita equivalence between k-graph algebras. Firstly, we define a C*-algebra generated by partial isometries, subject to some relations similar to the Cuntz-Krieger relations of a k-graph algebra. We then show that this C*algebra is isomorphic to a corner of a k-graph algebra. Since every k-graph algebra is trivially a corner of itself, then it follows that all k-graph algebras can be obtained this way. We show that this C*-algebra is universal and then prove an analogue of the Gauge Invariant Uniqueness Theorem for corners of k-graph algebras and then show a few applications of this theorem. Secondly, we define a way of generating a k-graph [C,d] from a category C and a functor δ : C → ℕk such that C is a full subcategory of [C,δ] and the degree map of [C,δ] is equal to d for any path that is the image of an element in C. This method is useful because it means one can define a k-graph from a category without needing to check if the factorisation property holds on that particular category. We then show some applications of generating a k-graph in this way. One application, in particular, is the k-graph analog of adding a tail to a directed graph. We also use this technique to generate some examples of the desingularisation of some k-graphs that are not row-finite.