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In mathematics, the Gelfand–Naimark theorem states that an arbitrary C *-algebra ''A'' is isometrically *-isomorphic to a C *-algebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C *-algebras since it established the possibility of considering a C *-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra. The Gelfand–Naimark representation π is the direct sum of representations π''f'' of ''A'' where ''f'' ranges over the set of pure states of A and π''f'' is the irreducible representation associated to ''f'' by the GNS construction. Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spaces ''H''''f'' by : π(''x'') is a bounded linear operator since it is the direct sum of a family of operators, each one having norm ≤ ||''x''||. Theorem. The Gelfand–Naimark representation of a C *-algebra is an isometric *-representation. It suffices to show the map π is injective, since for *-morphisms of C *-algebras injective implies isometric. Let ''x'' be a non-zero element of ''A''. By the Krein extension theorem for positive linear functionals, there is a state ''f'' on ''A'' such that ''f''(''z'') ≥ 0 for all non-negative z in ''A'' and ''f''(−''x'' * ''x'') < 0. Consider the GNS representation π''f'' with cyclic vector ξ. Since : it follows that π''f'' ≠ 0. Injectivity of π follows. The construction of Gelfand–Naimark ''representation'' depends only on the GNS construction and therefore it is meaningful for any Banach *-algebra ''A'' having an approximate identity. In general it will not be a faithful representation. The closure of the image of π(''A'') will be a C *-algebra of operators called the C *-enveloping algebra of ''A''. Equivalently, we can define the C *-enveloping algebra as follows: Define a real valued function on ''A'' by : as ''f'' ranges over pure states of ''A''. This is a semi-norm, which we refer to as the ''C * semi-norm'' of ''A''. The set I of elements of ''A'' whose semi-norm is 0 forms a two sided-ideal in ''A'' closed under involution. Thus the quotient vector space ''A'' / I is an involutive algebra and the norm : factors through a norm on ''A'' / I, which except for completeness, is a C * norm on ''A'' / I (these are sometimes called pre-C *-norms). Taking the completion of ''A'' / I relative to this pre-C *-norm produces a C *-algebra ''B''. By the Krein–Milman theorem one can show without too much difficulty that for ''x'' an element of the Banach *-algebra ''A'' having an approximate identity: : It follows that an equivalent form for the C * norm on ''A'' is to take the above supremum over all states. The universal construction is also used to define universal C *-algebras of isometries. Remark. The Gelfand representation or Gelfand isomorphism for a commutative C *-algebra with unit is an isometric *-isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of ''A'' with the weak * topology. ==See also== * GNS construction * Stinespring factorization theorem * Gelfand–Raikov theorem 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 theorem」の詳細全文を読む スポンサード リンク
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