On the Gel''fand representation of a commutative Banach algebra

In this paper the Gel'fand representation of a commutative Banach algebra is developed. The fundamental results are as follows. 1) Any complex commutative Banach algebra A is homomorphic to an algebra of continuous complex valued functions on a locally compact Hausdorff space. If A has an identity then the space is compact and in any case the functions vanish at infinity. The representation is norm decreasing. 2) If A is semi-simple the representation is an isomorphism. 3) If A is such that

x2

=

x

2 then the Gel'fand representation of A is isometric to A. Finally the Gel'fand representation is used to prove the Banach Stone Theorem and the essential uniqueness of the Stone-Cech compactification, and the Gel'fand representation of an element of L1(-00 ,00) is seen to be the Fourier transform of that element.