Gelfand–Mazur theorem

In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states:

A complex Banach algebra, with unit 1, in which every nonzero element is invertible, is isometrically isomorphic to the complex numbers.

In other words, the only complex Banach algebra that is a division algebra is the complex numbers C. This follows from the fact that, if A is a complex Banach algebra, the spectrum of an element a ∈ A is nonempty (which in turn is a consequence of the complex-analycity of the resolvent function). For every a ∈ A, there is some complex number λ such that λ1 − a is not invertible. By assumption, λ1 − a = 0. So a = λ · 1. This gives an isomorphism from A to C.

Actually, a stronger and harder theorem was proved first by Stanisław Mazur alone, but it was published in France without a proof, when the author refused the editor's request to shorten his already short proof. Mazur's theorem states that there are (up to isomorphism) exactly three real Banach division algebras: the fields of reals R, of complex numbers C, and the division algebra of quaternions H. Gelfand proved (independently) the easier, special, complex version a few years later, after Mazur. However, it was Gelfand's work which influenced the further progress in the area.

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