A groupoid approach to Cuntz-Krieger algebras

An irreducible $N\times N$-matrix $A$ over $\{0,1\}$ gives rise to a plethora of interesting mathematical objects. One can construct the \emph{one-sided topological Markov shift} $X_A$ on which the shift $\sigma_A$ acts. Alternatively, one can construct a $C^*$-algebra, the \emph{Cuntz-Krieger algebra} $\mathcal{A}$, or a groupoid $G_A$. In 2010, K. Matsumoto introduced the notion of \emph{continuous orbit equivalence} between topological Markov shifts and showed that this can be characterized in terms of $C^*$-isomorphisms between the Cuntz-Kriegers algebras which preserve a certain abelian $C^*$-subalgebra. In a recent paper, published in 2014, K. Matsumoto and H. Matui provided the final piece of a characterization of continuous orbit equivalence between Markov shifts. This includes the notions of $C^*$-algebras, groups and groupoids. In this thesis, we give a proof of this classification result. Along the way, we will see how to realize $\mathcal{A}$ as a groupoid $C^*$-algebra.