Gelfand–Naimark theorem
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 π of "A" where "f" ranges over the set of pure states  of A and π 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" 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 with cyclic vector ξ. Since it follows that π (x) ≠ 0, so π (x) ≠ 0, so π is injective. 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 (when "A" is not a C*-algebra) 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  formula_6 is an isometric *-isomorphism from formula_6 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.