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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.
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
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 (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.
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