The axioms of a complex Banach algebra were very happily chosen. They are simple enough to allow wide ranging fields of application,
notably in harmonic analysis, operator theory and function algebras.
At the same time they are tight enough to allow the development of a
rich collection of results, mainly through the interplay of the elementary
parts of the theories of analytic functions, rings, and Banach spaces.
Many of the theorems are things of great beauty, simple in statement,
surprising in content, and elegant in proof. We believe that some of
them deserve to be known by every mathematician.