In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebraA 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 Hilbert space analogue of 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 *-algebraA 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 spaceA / 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 *-algebraA 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, which in the commutative case are precisely the pure states, of A with the weak* topology.
![{\displaystyle \pi (x)[\bigoplus _{f}H_{f}]=\bigoplus _{f}\pi _{f}(x)H_{f}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95081fda4ca534e0dc15c765c9b4e604628c6c7)
![{\displaystyle {\begin{aligned}\|\pi _{f}(x)\xi \|^{2}&=\langle \pi _{f}(x)\xi \mid \pi _{f}(x)\xi \rangle =\langle \xi \mid \pi _{f}(x^{*})\pi _{f}(x)\xi \rangle \\[6pt]&=\langle \xi \mid \pi _{f}(x^{*}x)\xi \rangle =f(x^{*}x)>0,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd899623394b6032df1195d15b69ce29136865ad)
