Commutative Banach Algebras

Abstract

This chapter contains the Gelfand theory of commutative Banach algebras. The Gelfand spectrum and the Gelfand transform are introduced and analysed. The results are illustrated by various examples. In particular, it is explained in detail how under the Gelfand transformation piecewise continuous functions become continuous. Special attention is paid to finitely generated commutative Banach algebras and to the Banach algebra generated by a compact operator. The last two sections present applications to factorization of matrix functions and to Wiener-Hopf integral operators. The analysis starts with a study of multiplicative linear functionals.