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Gelfand-Naimark theorem

For a general C-star algebra A which is not necessarily commutative, the Gelfand-Naimark theorem states that A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. The isometric *-isomorphism constructed in the proof of this theorem is a direct sum of representations

of A where f ranges over the set of pure states of A and

is constructed from f by the GNS construction.

The Gelfand representation is a similar result which applies to commutative C*-algebras.

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