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.
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||.
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.
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.