We define the categorical cohomology of a k-graph Λ and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k-graph C * -algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k-graph C * -algebra is isomorphic to a twisted groupoid C * -algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k-graph C * -algebras are nuclear and belong to the bootstrap class.

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We associate to each row-finite directed graph E a universal Cuntz-Krieger C * -algebra C * (E), and study how the distribution of loops in E affects the structure of C * (E). We prove that C * (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the Cuntz-Krieger uniqueness theorem and give a characterisation of when C * (E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C * (E) is AF; if E has a loop, then C * (E) is purely infinite. , The ideal structure of groupoid crossed product C * -algebras, J. Operator Theory, 25 (1991), 3-36.

This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lowersemicontinuous gauge-invariant trace on C * (Λ) in terms of the existence of a faithful graph trace on Λ.Second, for k-graphs with faithful gauge invariant trace, we construct a smooth (k, ∞)summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the T k action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.2. k-Graph C * -Algebras 2.1. Higher-rank graphs and their C * -algebras. In this subsection we outline the basic notation and definitions of k-graphs and their C * -algebras. We refer the reader to [RSY] for a more thorough account.Higher-rank graphs. Throughout this paper, we regard N k as a monoid under pointwise addition. We denote the usual generators of N k by e 1 , . . . , e k , and for n ∈ N k and 1 ≤ i ≤ k, we denote the i th coordinate of n by n i ∈ N; so n = n i e i . For m, n ∈ N k , we write m ≤ n if m i ≤ n i for all i. By m < n, we mean m ≤ n and m = n. We use m ∨ n and m ∧ n to denote, respectively, the coordinate-wise maximum and coordinate-wise minimum of m and n; so that m ∧ n ≤ m, n ≤ m ∨ n and these are respectively the greatest lower bound and least upper bound of m, n in N k . Definition 2.1 (Kumjian-Pask [KP]). A graph of rank k or k-graph is a pair (Λ, d) consisting of a countable category Λ and a degree functor d : Λ → N k which satisfy the following factorisation property: if λ ∈ Mor(Λ) satisfies d(λ) = m + n, then there are unique morphisms µ, ν ∈ Mor(Λ) such that d(µ) = m, d(ν) = n, and λ = µ • ν.The factorisation property ensures (see [KP]) that the identity morphisms of Λ are precisely the morphisms of degree 0; that is {id o : o ∈ Obj(Λ)} = d −1 (0). This means that we may identify each object with its identity morphism, and we do this henceforth. This done, we can regard Λ as consisting only of its morphisms, and we write λ ∈ Λ to mean λ ∈ Mor(Λ).Since we are thinking of Λ as a kind of graph, we write r and s for the codomain and domain maps of Λ respectively. We refer to elements of Λ as paths, and to the paths of degree 0 (which correspond to the objects of Λ as above) as vertices. Extending these conventions, we refer to the elements of Λ with minimal nonzero degree (that is d −1 ({e 1 , . . . , e k }) as edges.

This paper explores the effect of various graphical constructions upon the associated graph C * -algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that outsplittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C * -algebras. We generalise the notion of a delay as defined in [D] to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C * -algebras. We provide examples which suggest that our results are the most general possible in the setting of the C * -algebras of arbitrary directed graphs.