C*- algebras were introduced, as natural abstractions of matrices acting on Hilbert spaces, by Murray and von Neumann in 1935 in order to explain certain physical observables of quantum mechanics. Today, they form an integral part of pure mathematics as they are able to generalize and improve on many of the notions we have for classic operators: eigenvalues, rank, projections etc. In this talk I will prove some spectral ("eigenvalue") theoretic results for Banach and C* - algebras.

I will start off by providing plenty of examples of C* and Banach algebras, as well as their spectral properties. Then I will move on to prove one of the most important spectral theorems for Banach algebras. If time permits, I will prove the Spectral Mapping Theorem and the existence of a continuous function calculus for self - adjoint operators of C* - algebras.

I aim to make the talk accessible for all attending, but knowledge of basic Functional Analysis will make you a little happier.