Gelfand-Mazur theorem
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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 λa - a is not invertible. By assumption, λa - a = 0. So a = λa ·1. This gives an isomorphism from A to C.
