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.

References

Rudin, Walter (1973), Functional analysis, Tata MacGraw-Hill .

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Strong (polar operator) Ultrastrong Weak (operator) Ultraweak Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Banach algebras	C-algebras Spectrum (C-algebra radius) Spectral theory

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure

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