# The Extension problem for Graph C*-Algebras – University of Copenhagen

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# The Extension problem for Graph C*-Algebras

Titel: The Extension problem for Graph $C^*$-Algebras
Abstract: To a directed graph $E$ we associate the graph $C^*$-algebra $C^*(E)$. This thesis addresses the extension problem for graph $C^*$-algebras: Given an extension (a short exact sequence) $0 \to C^*(E) \to A \to C^*(F) \to 0$ of $C^*$-algebras, where $C^*(E)$ and $C^*(F)$ are graph $C^*$-algebras, under which circumstances may it be deduced that $A$ is also a graph $C^*$-algebra? We investigate this question in the stable case such that $C^*(E)$ is a unique ideal in $A$. This is done by building a $K$-theoretic framework, which turns out to contain all the information necessary. First it is proven that the class of AF-algebras is closed under extensions, which is later applied to prove a special case of our main theorem. We then elaborate on the theory of graph $C^*$-algebras, proving among other things a dichotomy result stating that every simple graph $C^*$-algebra is either AF or purely infinite. Finally we show how to realize the ordered six-term exact sequence in $K$-theory induced by the above extension using graph $C^*$-algebras. We do this by showing that we can "glue" the two graphs $E$ and $F$ together to form a third graph with the required properties. In our main theorem we present necessary and sufficient $K$-theoretic criteria for $A$ to be a graph $C^*$-algebra.