Quaestiones Mathematicae

The AJOL site is currently undergoing a major upgrade, and there will temporarily be some restrictions to the available functionality.
-- Users will not be able to register or log in during this period.
-- Full text (PDF) downloads of Open Access journal articles will be available as always.
-- Full text (PDF) downloads of subscription based journal articles will NOT be available
We apologise for any inconvenience caused. Please check back soon, as we will revert to usual policy as soon as possible.

DOWNLOAD FULL TEXT Open Access  DOWNLOAD FULL TEXT Subscription or Fee Access

On Frobenius, Mazur, and Gelfand-Mazur theorems on division Algebras

VM Bogdan


The main purpose of this paper is to introduce the reader to the theory of normed division algebras. It is written with a graduate student of mathematics in mind, who only had exposure to Advanced Calculus and to an introduction to Functional Analysis. We establish the equivalence of the Mazur and Gelfand-Mazur theorems using only elementary properties of normed algebras. Mazur's theorem [19] states that every normed division algebra over the field of real numbers is isomorphic to either the field R of real numbers, the field C of complex numbers, or the non-commutative algebra Q of quaternions. Gelfand [15] proved that every normed division algebra over the field C is isomorphic to C. He named this theorem, which is fundamental for the development of the theory of Banach Algebras, the Gelfand-Mazur theorem. An analysis of Mazur's proof (see [27], p. 18–23) shows that he proved that theorem firstly for algebras over C (using the theory of Analytic Functions) and then generalized the proof to the case of algebras over R (using the theory of Harmonic Functions.) We prove the theorems that are essential for the development in detail. These include the construction of the algebra of quaternions and the theorems of Frobenius [14], Arens [1], and Stone [26].

Keywords: Mazur theorem, Gelfand-Mazur theorem, Frobenius theorem, Banach algebras, normed algebras, division algebras, topological algebras, topological fields, topological rings

Quaestiones Mathematicae 29(2006), 171–209

Full Text:

No subscription journal articles available during site upgrade.
AJOL African Journals Online