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 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 "λ_a - a" is not invertible. By assumption, "λ_a - a" = 0. So "a" = "λ_a ·"1. This gives an isomorphism from "A" to C.