Teoremi di struttura e unicità delle graph C*-algebras

Graph C*-algebras are operator algebras arising from directed graphs via a construction developed by Cuntz and Krieger in 1980. The most significant reason to study these algebras is to visualise operator algebras and some of their properties looking at their associated graphs. The aim of this thesis is to introduce the general theory of graph C*-algebras and unfold its fundamental theorems. The main results state sufficient conditions the graph must satisfy to guarantee uniqueness for its arising graph C*-algebra. Secondly, some properties of a graph C*-algebra are characterised by graph features. In particular, the first two chapters outline the indispensable background theory of C*-algebras, including the GNS construction. The third chapter includes some basic facts in the Hilbert space theory which are necessary afterwards. The fourth chapter consists of the Cuntz-Krieger construction of a graph C*-algebra given a directed graph and its universal property. In the fifth chapter complete proofs of the gauge-invariant and Cuntz-Krieger theorems are presented, and discussed subsequently with examples. The sixth chapter is about the structure of graph C*-algebras, at the end a characterisation of closed ideals and simplicity is given. Finally, in the seventh chapter the K-theory for C*-algebras is introduced, and then the special case of K-theory for graph C*-algebra is shown.