A Categorical Study of C*-correspondences and Associated Cuntz-Pimsner Algebras

This thesis presents a categorical approach to recover and extend some well-known results from the theory of $C^*$-correspondences. First, a detailed study on \emph{the enchilada category} is given. In this category, the objects are $C^*$-algebras, and morphisms are isomorphism classes of $C^*$-correspondences. In the \emph{equivariant enchilada category} $\mathcal{A}(C)$, all objects and morphisms are equipped with a locally compact group action satisfying certain conditions. These categories were used by Echterhoff, Kaliszweski, Quigg, and Raeburn for a study regarding a very fundamental tool to the representation theory: imprimitivity theorems. This work contains a construction of exact sequences in enchilada categories. One of the main results is that the reduced-crossed-product functor defined from $\mathcal{A}(C)$ to the enchilada category is not exact. The motivation was to determine whether one can have a better understanding of the Baum-Connes conjecture. Along the way numerous results are proven, showing that the enchilada category is rather strange. The next main study regards the functoriality of Cuntz-Pimsner algebras. A construction of a functor that maps $C^*$-correspondences to their Cuntz-Pimsner algebras is presented. The objects in the domain category are $C^*$-correspondences, and the morphisms are the isomorphism classes of $C^*$-correspondences satisfying certain conditions. Applications include a generalization of the well-known result of Muhly and Solel: Morita equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras; as well as a generalization of the result of Kakariadis and Katsoulis: Regular shift equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras.