## The Gelfand-Naimark theorem for commutative Banach star algebras

##### Summary

We start by studying properties of several kinds of algebras, taking a look at the spectrum, ideals and abelian algebras. Then we prove the Gelfand-Naimark Theorem for commutative Banach star algebras (also known as abelian C*-algebras). After that, we prove some basic theorems about Riemann integration of Banach valued functions. We then study some applications of the Gelfand-Naimark Theorem where we start by studying the functional calculus, in particular the Riesz functional calculus and its extension to C*-algebras. We then take a look at positive elements, representations of C*-algebras and in particular the Gelfand-Naimark-Segal construction. Lastly, we study spectral measures and, using representations, we prove the spectral theorem for bounded normal operators on a Hilbert space.