Searcharxiv⌕ Search

arXiv subjects

Mark Tomforde

Publications and source records attributed to Mark Tomforde.

47 records · Page 3Linked to original sources
arXiv2007

Uniqueness Theorems and Ideal Structure for Leavitt Path Algebras

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).

math.OA↗
arXiv2005

Topological Quivers

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.

math.OA↗
arXiv2004

Adding tails to C*-correspondences

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.

math.OA↗
arXiv2004

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.

math.OA↗
arXiv2003

Simplicity of ultragraph algebras

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.

math.OA↗
arXiv2003

Computing Ext for graph algebras

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.

math.OA↗
arXiv2003

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 F is the graph formed by adding a head to each vertex of G, then C*(F) is the stabilization of C*(G).

math.OA↗
arXiv2002

The ordered K_0-group of a graph C*-algebra

We calculate the ordered K_0-group of a graph C*-algebra and mention applications of this result to AF-algebras, states on the K_0-group of a graph algebra, and tracial states of graph algebras.

math.OA↗
arXiv2001

Ext classes and embeddings for C*-algebras of graphs with sinks

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.

math.OA↗
arXiv2001

A unified approach to Exel-Laca algebras and C*-algebras associated to graphs

We define an ultragraph, which is a generalization of a directed graph, and describe how to associate a C*-algebra to it. We show that the class of ultragraph algebras contains the C*-algebras of graphs as well as the Exel-Laca algebras. We also show that many of the techniques used for graph algebras can be applied to ultragraph algebras and that the ultragraph provides a useful tool for analyzing Exel-Laca algebras. Our results include versions of the Cuntz-Krieger Uniqueness Theorem and the Gauge-Invariant Uniqueness Theorem for ultragraph algebras.

math.OA↗