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 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
\pi(x)[oplusfHf]=oplusf\pif(x)Hf.
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
\begin{align} \|\pif(x)\xi\|2&=\langle\pif(x)\xi\mid\pif(x)\xi\rangle =\langle\xi\mid
| *) | |
| \pi | |
| f(x |
\pif(x)\xi\rangle\\[6pt] &=\langle\xi\mid
| * | |
| \pi | |
| f(x |
x)\xi\rangle=f(x*x)>0, \end{align}
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
| *} | |
| \|x\| | |
| \operatorname{C |
=\supf\sqrt{f(x*x)}
\| ⋅
| *} | |
| \| | |
| \operatorname{C |
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:
\supf(A)}f(x*x)=\supf(A)}f(x*x).
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
A
A
{\operatorname{C}*}