  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.
