Description
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.
Details
Title
- A Categorical Study of C*-correspondences and Associated Cuntz-Pimsner Algebras
Contributors
- ERYUZLU, MENEVSE (Author)
- Quigg, John (Thesis advisor)
- Kalszewski, Steven (Committee member)
- Spielberg, Jack (Committee member)
- Williams, Dana (Committee member)
- Larsen, Nadia (Committee member)
- Arizona State University (Publisher)
Date Created
The date the item was original created (prior to any relationship with the ASU Digital Repositories.)
2021
Subjects
Resource Type
Collections this item is in
Note
- Partial requirement for: Ph.D., Arizona State University, 2021
- Field of study: Mathematics