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Gel'fand-naimark TheoremIn 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 a significant point in the development of the theory of C*-algebras in the early 1940s since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an algebra of operators. 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 -
Note that 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. The construction of Gelfand-Naimark representation depends only on the GNS construction and therefore it is meaningful for any B*-algebra A having an approximate identity. In general 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 B*-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 is an isometric *-isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals of A with the weak* topology. Reference - I. M. Gelfand and M. A. Naimark, On the imbeding of normed rings into the ring of operators on aHilbert space, Math. Sbornik, vol 12, 1943.
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