Mark Tomforde

Isomorphism and Morita equivalence of graph algebras

For any countable graph E, we investigate the relationship between the Leavitt path algebra L ℂ (E) and the graph C * -algebra C * (E). For graphs E and F, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between L ℂ (E) and L ℂ (F). We prove that in certain situations isomorphisms between L ℂ (E) and L ℂ (F) yield *-isomorphisms between the corresponding C * -algebras C * (E) and C * (F). Conversely, we show that *-isomorphisms between C * (E) and C * (F) produce isomorphisms between L ℂ (E) and L ℂ (F) in specific cases. The relationship between Leavitt path algebras and graph C * -algebras is also explored in the context of Morita equivalence.

Operator system structures on ordered spaces

Given an Archimedean order unit space (V,V^+,e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system S is completely boundedly isomorphic to either OMIN(S) or to OMAX(S). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(M_n) to OMAX(M_m) if and only if it is entanglement breaking.

Stability of C^*-algebras associated to graphs

We characterize stability of graph C*-algebras by giving five conditions equivalent to their stability. We also show that if G is a graph with no sources, then C*(G) is stable if and only if each vertex in G can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph C*-algebra. Specifically, if G is a graph and G is the graph formed by adding a head to each vertex of G, then C*(G) is the stabilization of C*(G); that is, C*(G) ≅ C*(G) ⊗ K.

INDEX MAPS IN THE K-THEORY OF GRAPH ALGEBRAS

Let C � (E) be the graph C � -algebra associated to a graph E and let J be a gauge-invariant ideal in C � (E). We compute the cyclic six-term exact sequence in K-theory associated to the extension 0 ! J ! C � (E) ! C � (E)/J ! 0 in terms of the adjacency matrix associated to E. The ordered six- term exact sequence is a complete stable isomorphism invariant for se- veral classes of graph C � -algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences comprise complete invariants. Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.

GRAPH ALGEBRAS, EXEL-LACA ALGEBRAS, AND ULTRAGRAPH ALGEBRAS COINCIDE UP TO MORITA EQUIVALENCE

Abstract We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph we construct a directed graph E such that is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals.

The ranges of -theoretic invariants for nonsimple graph algebras

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K-groups by splicing together smaller graphs whose C*-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigations have more than one ideal and where there are currently no relevant classification theories available.

Classification theorems for the C*-algebras of graphs with sinks

We consider graphs E which have been obtained by adding one or more sinks to a fixed directed graph G . We classify the C * -algebra of E up to a very strong equivalence relation, which insists, loosely speaking, that C * ( G ) is kept fixed. The main invariants are vectors W E : G 0 → ℕ which describe how the sinks are attached to G ; more precisely, the invariants are the classes of the W E in the cokernel of the map A – I , where A is the adjacency matrix of the graph G .

Classification of Graph Algebras: A Selective Survey

This survey reports on current progress of programs to classify graph C∗-algebras and Leavitt path algebras up to Morita equivalence using K-theory. Beginning with an overview and some history, we trace the development of the classification of simple and nonsimple graph C∗-algebras and state theorems summarizing the current status of these efforts. We then discuss the much more nascent efforts to classify Leavitt path algebras, and we describe the current status of these efforts as well as outline current impediments that must be solved for this classification program to progress. In particular, we give two specific open problems that must be addressed in order to identify the correct K-theoretic invariant for classification of simple Leavitt path algebras, and we discuss the significance of various possible outcomes to these open problems.

The prime spectrum and primitive ideal space of a graph C ∗ -algebra

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.