nLab
Gelfand-Mazur theorem

The Gel’fand–Mazur theorem states: the only complex Banach algebra which is also a field is the algebra of complex numbers \mathbb{C}.

The proof is a very simple consequence of the spectral theory of elements in a unital complex Banach algebra. It is a basic result of spectral theory that the spectrum sp(A)sp(A) of any element aa in a Banach algebra AA (which is by definition the set of complex numbers λ\lambda such that aλ1a- \lambda 1 is not invertible) is a nonempty compact subset of \mathbb{C}. Now if the algebra is a field (or even a skewfield) then the only noninvertible element is 00, hence every point in the spectrum of an arbitrary element aAa\in A provides λ\lambda such that λ1=a\lambda 1 = a. Therefore the algebra can be identified with a unital complex subalgebra of \mathbb{C}, hence it is \mathbb{C}.

Revised on November 15, 2009 00:32:37 by Toby Bartels (173.60.119.197)