Gelfand-Naimark theorem
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 πf of A where f ranges over the set of pure states of A and πf 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 Hf 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 πf with cyclic vector ξ. Since
it follows that πf ≠ 0. Injectivity of π follows.
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