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:

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*1 − *a* is not invertible. By assumption, *λ*1 − *a* = 0. So *a* = *λ · *1. This gives an isomorphism from *A* to **C**.

Actually, a stronger and harder theorem was proved first by Stanislaw 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. Gelfand has created a whole theory.

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- 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

Actually, a stronger and harder theorem was proved first by Stanislaw 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

- .

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