On The Higher Analytical Index

Hans, Satwata (2022) On The Higher Analytical Index. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS dissertation of Satwata Hans (17MS059)) 17MS059_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this project, we study an interesting sub-class of operators called Fredholm Operators. The Fredholm Operators coupled with the Fredholm Index map provide a lot of information about the spaces we are trying to study, and has a natural connection to K-Theory. The primary goal of this project is to give a complete classification of the K-Theory of a unital C∗ algebra A in terms of a higher index on generalized Fredholm operators. To be precise, we will look at two important theorems relating Fredholm Theory and K-Theory. One of them is called the Atiyah-Jänich Theorem; it provides a classification of the K-Theory of any compact Hausdorff space in terms of Fredholm operators on a separable Hilbert space H. The other result by J.A. Mingo provides a complete description of the K-Theory of a unital C∗ Algebra in terms of generalized Fredholm Operators on a Hilbert C∗ module. In between that, we will look at an application of Fredholm Theory and K-Theory to the idempotent conjecture in algebra by Irving Kaplansky.

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