# Gelfand–Naimark theorem

__: Mathematics theorem in functional analysis__

**Short description**

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_{f}
of *A* where *f* ranges over the set of pure states of A and π_{f} 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*_{f} by

- [math]\displaystyle{ \pi(x) [\bigoplus_{f} H_f] = \bigoplus_{f} \pi_f(x)H_f. }[/math]

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_{f} with cyclic vector ξ. Since

- [math]\displaystyle{ \begin{align} \|\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) \gt 0, \end{align} }[/math]

it follows that π_{f} (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

- [math]\displaystyle{ \|x\|_{\operatorname{C}^*} = \sup_f \sqrt{f(x^* x)} }[/math]

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

- [math]\displaystyle{ \| \cdot \|_{\operatorname{C}^*} }[/math]

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:

- [math]\displaystyle{ \sup_{f \in \operatorname{State}(A)} f(x^*x) = \sup_{f \in \operatorname{PureState}(A)} f(x^*x). }[/math]

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 [math]\displaystyle{ A }[/math] is an isometric *-isomorphism from [math]\displaystyle{ A }[/math] 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.

## See also

- GNS construction
- Stinespring factorization theorem
- Gelfand–Raikov theorem
- Koopman operator
- Tannaka–Krein duality

## References

- I. M. Gelfand, M. A. Naimark (1943). "On the imbedding of normed rings into the ring of operators on a Hilbert space".
*Mat. Sbornik***12**(2): 197–217. http://mi.mathnet.ru/eng/msb6155. (also available from Google Books) - Dixmier, Jacques (1969).
*Les C*-algèbres et leurs représentations*. Gauthier-Villars. ISBN 0-7204-0762-1. https://archive.org/details/calgebras0000dixm., also available in English from North Holland press, see in particular sections 2.6 and 2.7. - Eisner, Tanja; Farkas, Bálint; Haase, Markus; Nagel, Rainer (2015). "The [math]\displaystyle{ {\operatorname{C}^*} }[/math]-Algebra C(K) and the Koopman Operator".
*Operator Theoretic Aspects of Ergodic Theory*. Springer. pp. 45–70. doi:10.1007/978-3-319-16898-2_4. ISBN 978-3-319-16897-5.

Original source: https://en.wikipedia.org/wiki/Gelfand–Naimark theorem.
Read more |