### Abstract:

The aim of this dissertation will be an investigation into a classical result which asserts
the uniqueness of the norm topology on a semi-simple Banach algebra.
For a commutative semi-simple Banach algebra, say A, it is relatively simple matter, with
the aid of the Closed Graph Theorem, to show that all Banach algebra norms on A must
be equivalent. The same result for non-commutative Banach algebras was conjectured by
I. Kaplansky in the 1950’s and solved more then a decade later, in 1967, by B E Johnson.
However, Johnson’s proof was difficult and relied heavily on representation theory. As a
result, the problem remained unsolved for the more difficult situation of Jordan Banach
algebras. Fifteen years later in 1982, B. Aupetit succeeded in proving Johnson’s result,
using a subharmonic method that was independent of algebra representations. Moreover
he could, using these techniques, also settle the problem in the Jordan Banach algebra
case. A while later, in 1989, T. Ransford provided a shorter algebraic proof of Johnson’s
result using the well-known spectral radius formula. This dissertation will be a comparative
study of the three different approaches on the problem for Banach algebras.