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The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the theory of locally compact topological groups. It states that a locally compact group is completely determined by its (possibly infinity dimensional) unitary representations. More precisely, the points of a locally compact topological group ''G'' are separated by its irreducible unitary representations. In other words, for any two group elements ''g'',''h'' ∈ ''G'' there exist an irreducible unitary representation ''ρ'' : ''G'' → U(''H'') such that ρ(''g'') ≠ ρ(''h''). It then follows from the Stone–Weierstrass theorem that on every compact subset of the group, the continuous functions defined by <''e''i'''', ρ(''g'')''e''j''''> with ''e''''i'' orthonormal basis vectors in ''H'' (the matrix coefficients), are dense in the space of continuous functions. The theorem was first published in 1943.〔(И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943), 301–316 ), (I. Gelfand, D. Raikov, "Irreducible unitary representations of locally bicompact groups", Rec. Math. N.S., 13(55):2–3 (1943), 301–316)〕 〔(Yoshizawa, Hisaaki. "Unitary representations of locally compact groups. Reproduction of Gelfand–Raikov's theorem." Osaka Mathematical Journal 1.1 (1949): 81–89 ).〕 ==See also== * Gelfand–Naimark theorem * Representation theory 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–Raikov theorem」の詳細全文を読む スポンサード リンク
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