3 years ago

# Gelfand–Mazur Theorems in normed algebras: A survey

S.j. Bhatt, S.h. Kulkarni

Publication date: June 2018

Source: Expositiones Mathematicae, Volume 36, Issue 2

Author(s): S.J. Bhatt, S.H. Kulkarni

##### 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.

DOI: S0723086917300567

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