Knopp, Paul J.2022-01-272022-01-27196613683883https://hdl.handle.net/10657/8623In 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 thatx2=x2 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.application/pdfenThis item is protected by copyright but is made available here under a claim of fair use (17 U.S.C. §107) for non-profit research and educational purposes. Users of this work assume the responsibility for determining copyright status prior to reusing, publishing, or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires express permission of the copyright holder.Thesisreformatted digital