ABSTRACT
This chapter contains a brief exposition of that part of Banach algebra theory that will be needed in the rest of this book, including the spectral theorem for commutative C* algebras. Although these topics are not part of harmonic analysis as such, the Gelfand transform and the spectral theorem are embodiments of ideas that are also central to harmonic analysis: the conversion of operators into more transparent forms and the decomposition of operators into simpler pieces.
