Extensions of graph C*-algebras

Ph.D. Thesis

ABSTRACT: We consider C*-algebras associated to row-finite (directed) graphs and examine the effect that adding a sink to the graph has on the associated C*-algebra. In Chapter 2 we give a precise definition of how a sink may be added to a graph, and discuss a notion of equivalence of C*-algebras if this is done in two different ways. We also define operations that may be performed on these graphs and then use these operations to determine equivalence of the associated C*-algebras in certain circumstances. In Chapters 3 and 4 we discuss the Ext functor and show that adding a sink to a graph G determines an element c of Ext (C*(G)). With this in mind, we construct an isomorphism w : Ext (C*(G)) -> coker (A-I), where A is the vertex matrix of G. We also show that the value that w assigns to c is the class of a vector describing how the sink was added to G. In Chapter 5 we use this isomorphism to strengthen the results of Chapter 2. In particular, if two graphs are formed by adding a sink to G, then we give conditions for their associated C*-algebras to be equivalent in terms of the vectors describing how the sinks were added.

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The material in this thesis has been rewritten for publication in the following three papers.

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