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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 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 ''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 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== * . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gelfand–Mazur theorem」の詳細全文を読む スポンサード リンク
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