We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

Gene Abrams Mark Tomforde

The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.

Gene Abrams Efren Ruiz Mark Tomforde

We consider directed graphs E which have been obtained by adding a sink to a fixed graph G. We associate an element of Ext(C*(G)) to each such E, and show that the classes of two such graphs are equal in Ext(C*(G)) if and only if the associated C*-algebra of one can be embedded as a full corner in the C*-algebra of the other in a particular way. If every loop in G has an exit, then we are able to use this result to generalize some of the classification theorems of Raeburn, Tomforde, and Williams for C*-algebras of graphs with sinks.

Mark Tomforde

We define a notion of strong shift equivalence for $C^*$-correspondences and show that strong shift equivalent $C^*$-correspondences have strongly Morita equivalent Cuntz-Pimsner algebras. Our analysis extends the fact that strong shift equivalent square matrices with non-negative integer entries give stably isomorphic Cuntz-Krieger algebras.

Paul Muhly David Pask Mark Tomforde

We give a complete $K$-theoretical description of when an extension of two simple graph $C^{*}$-algebras is again a graph $C^{*}$-algebra.

Søren Eilers James Gabe Takeshi Katsura Efren Ruiz Mark Tomforde

For a row-finite graph G with no sinks and in which every loop has an exit, we construct an isomorphism between Ext(C*(G)) and coker(A-I), where A is the vertex matrix of G. If c is the class in Ext(C*(G)) associated to a graph obtained by attaching a sink to G, then this isomorphism maps c to the class of a vector which describes how the sink was added. We conclude with an application in which we use this isomorphism to produce an example of a row-finite transitive graph with no sinks whose associated C*-algebra is not semiprojective.

Mark Tomforde

In this paper we analyze the structure of C*-algebras associated to ultragraphs, which are generalizations of directed graphs. We characterize the simple ultragraph algebras as well as deduce necessary and sufficient conditions for an ultragraph algebra to be purely infinite and to be AF. Using these techniques we also produce an example of an ultragraph algebra which is neither a graph algebra nor an Exel-Laca algebra. We conclude by proving that the C*-algebras of ultragraphs with no sinks are Cuntz-Pimsner algebras.

Mark Tomforde

We describe a method of adding tails to C*-correspondences which generalizes the process used in the study of graph C*-algebras. We show how this technique can be used to extend results for augmented Cuntz-Pimsner algebras to C*-algebras associated to general C*-correspondences, and as an application we prove a gauge-invariant uniqueness theorem for these algebras. We also define a notion of relative graph C*-algebras and show that properties of these C*-algebras can provide insight and motivation for results about relative Cuntz-Pimsner algebras.

Paul S. Muhly Mark Tomforde

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C*-correspondence, and from this correspondence one may construct a Cuntz-Pimsner algebra C*(Q). In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of C*(Q) can be determined from Q. We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the Gauge-Invariant Uniqueness theorem, the Cuntz-Krieger Uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.

Paul S. Muhly Mark Tomforde

We prove Leavitt path algebra versions of the two uniqueness theorems of graph C*-algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the fact that for any graph E the Leavitt path algebra $L_\mathbb{C}(E)$ embeds as a dense *-subalgebra of the graph C*-algebra C*(E). This embedding has consequences for graph C*-algebras, and we discuss how we obtain new information concerning the construction of C*(E).

Mark Tomforde