Gelfand–Naimark theorem

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

: \pi(x) bigoplus_ H_f= \bigoplus_ \pi_f(x)H_f.

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

:
\begin
\, \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 \\ pt& = \langle \xi \mid \pi_f(x^* x) \xi \rangle= f(x^* x) > 0,
\end

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
:$\, x\, _ = \sup_f \sqrt$
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

:$\, \cdot \, _$

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:
:$\sup_ f(x^*x) = \sup_ f(x^*x).$
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 $A$ is an isometric *-isomorphism from $A$ 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
* Tannaka–Krein duality

References

* (als
available from Google Books

* , also available in English from North Holland press, see in particular sections 2.6 and 2.7.

{{DEFAULTSORT:Gelfand-Naimark theorem
Operator theory
Theorems in functional analysis
C*-algebras