$C^*$-algebras: The (Quantum) Path Less Traveled

Abstract

In this project, we will develop the theory of Banach algebras and prove two celebrated theorems, the Gelfand representation theorem and the GKZ theorem. We will then proceed to develop the theory of $C^*$-algebras and prove the Gelfand-Naimark theorem and the GNS theorem. Finally, we will show how they can be used to give an alternative formulation of quantum mechanics.