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Concept# Gelfand–Naimark theorem

Summary

In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra 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.
Details
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
: \pi(x) [\bigoplus_{f} H_f] = \bigoplus_{f} \pi_f(x)H_f.
π(x) is a bounded linear operator since it is the direct sum of a family of operators, each one having norm ≤ ||x

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