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In functional analysis, a discipline within mathematics, given a C *-algebra ''A'', the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of ''A'' and certain linear functionals on ''A'' (called ''states''). The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal. == States and representations == A *-representation of a C *-algebra ''A'' on a Hilbert space ''H'' is a mapping π from ''A'' into the algebra of bounded operators on ''H'' such that * π is a ring homomorphism which carries involution on ''A'' into involution on operators * π is nondegenerate, that is the space of vectors π(''x'') ξ is dense as ''x'' ranges through ''A'' and ξ ranges through ''H''. Note that if ''A'' has an identity, nondegeneracy means exactly π is unit-preserving, i.e. π maps the identity of ''A'' to the identity operator on ''H''. A state on C *-algebra ''A'' is a positive linear functional ''f'' of norm 1. If ''A'' has a multiplicative unit element this condition is equivalent to ''f''(1) = 1. For a representation π of a C *-algebra ''A'' on a Hilbert space ''H'', an element ξ is called a cyclic vector if the set of vectors : is norm dense in ''H'', in which case π is called a cyclic representation. Any non-zero vector of an irreducible representation is cyclic. However, non-zero vectors in a cyclic representation may fail to be cyclic. 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–Naimark–Segal construction」の詳細全文を読む スポンサード リンク
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