Gelfand–Mazur Theorems in normed algebras: A survey

Abstract

The Gelfand–Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers R, the complex numbers C or the quaternions H; (Complex version) Every complex normed division algebra is isometrically isomorphic to C. This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues.