Fredholm Operators over $C^*$-Hilbert Modules

The content here presented was produced during the last academic year of my bachelor’s degree in Mathematics at the Institute of Mathematics and Statistics of the University of São Paulo (IME-USP).

Context

The present document builds upon Ruy Exel’s paper “A Fredholm Operator approach to Morita Equivalence”, in a closer look to details. In it, the theory of generalized Fredholm operators between Hilbert $C^\star$-modules is developed, and a generalization of the Brown, Green and Rieffel theorem concerning induced K-groups on equivalent Morita-Rieffel $C^*$-algebras is demonstrated.

Here, I make sure to include all definitions and theorems necessary to understand the machinery developed by Exel in that article.

Content

• $K$-theory of Banach Algebras
• Hilbert and Finite-rank $C^*$-Modules
• Generalised Fredholm Operators
• Application to Morita Equivalence

Compiling

You can generate the present report with the following command:

source ./compiler.bash