Iain Raeburn

Higher-rank graphs and their C*-algebras

We consider the higher-rank graphs introduced by Kumjian and Pask as models for higherrank Cuntz–Krieger algebras. We describe a variant of the Cuntz–Krieger relations which applies to graphs with sources, and describe a local convexity condition which characterizes the higher-rank graphs that admit a non-trivial Cuntz–Krieger family. We then prove versions of the uniqueness theorems and classifications of ideals for the C∗-algebras generated by Cuntz–Krieger families.

Cuntz-Krieger algebras of infinite graphs and matrices

We show that the Cuntz-Krieger algebras of infinite graphs and infinite {0,1}-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their K-theory. Since the finite approximating graphs have sinks, we have to calculate the K-theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.

Representations of crossed products by coactions and principal bundles

The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on C*-algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on C*-algebras, and to apply these ideas to classes of coactions termed pointwise unitary and locally unitary to obtain a complete realization of the isomorphism theory of locally trivial principal G-bundles in this context. We are also able to obtain all (Cartan) principal G-bundles in this context, but their isomorphism theory remains elusive.

Pull-backs of *-algebras and crossed products by certain diagonal actions

Let G be a locally compact group and p: Q -T a principal G-bundle. If A is a C*-algebra with primitive ideal space T, the pull-back p*A of A along p is the balanced tensor product Co((Q) ?C(T) A. If ,3: G -Aut A consists of C(T)-module automorphisms, and -y: G -Aut Co (Q) is the natural action, then the automorphism group y 0X 3 of Co(Q) 0 A respects the balancing and induces the diagonal action p*,3 of G on p*A. We discuss some examples of such actions and study the crossed product p* A x p G. We suggest a substitute D for the fixed-point algebra, prove p*A x G is strongly Morita equivalent to D, and investigate the structure of D in various cases. In particular, we ask when D is strongly Morita equivalent to A-sometimes, but by no means always-and investigate the case where A has continuous trace. Let B be a C*-algebra and G a locally compact group acting on B as a strongly continuous automorphism group a. Our goal here is to study the crossed product C*-algebra B x , G for two classes of diagonal actions for which the induced action of G on B is free. The first class includes actions of the form -y 0X3 on B Co (Q) 0 A, where p: Q -* T is a principal G-bundle, -y is the dual action of G on Co (Q), and 3: G -* Aut A is an action of G on another C*-algebra A. We also consider diagonal actions on algebras which are the pull-backs of another algebra A along a principal bundle p: Q -* T: if A is a C*-algebra with primitive ideal space T, then the pull-back p*A is the balanced tensor product Co(Q) ?Cb(T) A. When 3: G -* Aut A consists of C(T)-module automorphisms, the product action ty 0 /3 preserves the balancing, and the diagonal action p*3 is, by definition, the induced action on p* A. In general, if f: X -* Y and q: PrimA -* Y are continuous, then Cb(Y) acts on Co(X) by composition with f, and on A by composition with q and the DaunsHofmann theorem. We can therefore define the pull-back f*A of A along f as the C*-algebraic tensor product Co(X) ?Cb(y) A. The reason for the name is that when A is the algebra of sections of some C*-bundle E over Y, there is a natural isomorphism of f*A onto the algebra of sections of the pull-back f*E. In ?1 we discuss this and other basic properties of pull-backs and give some evidence to show they are likely to be of interest. In particular, we show that if G is abelian and a: G -* Aut A is locally unitary in the sense of [18], then the crossed product Received by the editors October 20, 1983 and, in revised form, February 15, 1984. 1980 Mathematics Subject Classification. Primary 46L40, 46L55.

The *-algebras of infinite graphs

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Kumjian-Pask algebras of higher-rank graphs

We introduce higher-rank analogues of the Leavitt path algebras, which we call the Kumjian-Pask algebras. We prove graded and Cuntz-Krieger uniqueness theorems for these algebras, and analyze their ideal structure.

Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Results of Fowler and Sims show that every k-graph is completely deter- mined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated to a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomor- phism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graphis isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterisation, originally due to Robertson and Sims, of simplicity of the C � -algebra of a row-finite k-graph with no sources.

Exel's crossed product and relative Cuntz–Pimsner algebras

We consider Exels new construction of a crossed product of a C ∗ -algebra A by an endomorphism α. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz-Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from A into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product. In this paper, we re-examine Exels crossed product, denoted A� α,LN ,a nd identify a family of representations for which A� α,LN is universal. We then show that A� α,LN can be realised as a relative Cuntz-Pimsner algebra as in (6, 11), and use known results for relative Cuntz-Pimsner algebras to study A� α,LN. In particular, we identify conditions which ensure that the canonical map A → A� α,LN is injective, thus answering a question raised by Exel in (3), and partially answered by him in

EQUILIBRIUM STATES ON THE CUNTZ-PIMSNER ALGEBRAS OF SELF-SIMILAR ACTIONS

We consider a family of Cuntz–Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz–Pimsner algebra; if the self-similar group is contracting, then the Cuntz–Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.

Exel's crossed product for non-unital C ∗ -algebras

We consider a family of dynamical systems (A ,α ,L) in which α is an endomorphism of a C ∗ -algebra A and L is a transfer operator for α. We extend Exel’s construction of a crossed product to cover non-unital algebras A, and show that the C ∗ -algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.