Quasi-free actions on graph algebras: KMS states and the structure of crossed products

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University of New Brunswick
This dissertation deals with the study of C*-dynamical systems. A C*-dynamical system consists of a space called a C*-algebra along with an action of a group on it. The systems that are the object of study here are ones that consist of C*-algebras obtained from directed graphs, along with actions, known as quasi-free actions, which depend on a labeling map. In physics, C*-dynamical systems are used to model physical systems. In these models, the states of the system are described by certain linear functionals on the algebra. The main results of this dissertation deal with the study of Kubo-Martin-Schwinger (KMS) states. These are the functionals that describe the equilibrium states of the physical system. Within this dissertation, a complete characterization of KMS states is given for C*-dynamical systems where the C*-algebra comes from a finite graph and the real action is quasi-free. This characterization provides a framework to generalize resultsfor a specific example of a quasi-free action known as the gauge action. More specifically, an explicit construction of all KMS states above a certain critical inverse temperature is provided, as well as a complete description of the KMS states when the corresponding graph has a certain strongly connected subgraph. In the study of C*-dynamical systems a certain algebra, called a crossed product, is used to describe the representations of the system. The structure of crossed products of graph algebras by quasi-free actions is also investigated in this thesis. For certain graph algebras, it is shown that some of the crossed products have nice inductive limit structures, extending known results for crossed products of Cuntz algebras by quasi-free actions.