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Gelfand–Naimark theorem

{{Short description|Mathematics theorem in functional analysis}}

{{Distinguish|Gelfond–Schneider 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 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) [\bigoplus_{f} H_f] = \bigoplus_{f} \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 πf with cyclic vector ξ. Since

:

\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) > 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}^*} = \sup_f \sqrt{f(x^* x)}

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 \|_{\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:

: \sup_{f \in \operatorname{State}(A)} f(x^*x) = \sup_{f \in \operatorname{PureState}(A)} 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

References

{{Reflist}}

  • {{cite journal |author=I. M. Gelfand, M. A. Naimark |title=On the imbedding of normed rings into the ring of operators on a Hilbert space |journal=Mat. Sbornik |volume=12 |issue=2 |year=1943 |pages=197–217 |url=http://mi.mathnet.ru/eng/msb6155}} (also [https://books.google.com/books?id=DYCUp0JYU6sC&pg=PA3 available from Google Books])
  • {{cite book |first=Jacques|last=Dixmier|author-link=Jacques Dixmier|title=Les C*-algèbres et leurs représentations|publisher=Gauthier-Villars|year=1969|isbn=0-7204-0762-1|url-access=registration|url=https://archive.org/details/calgebras0000dixm}}, also available in English from North Holland press, see in particular sections 2.6 and 2.7.
  • {{cite book |first=Tanja |last=Eisner |first2=Bálint |last2=Farkas |first3=Markus |last3=Haase |first4=Rainer |last4=Nagel |year=2015 |title=Operator Theoretic Aspects of Ergodic Theory |location= |publisher=Springer |isbn=978-3-319-16897-5 |doi=10.1007/978-3-319-16898-2_4 |chapter=The {\operatorname{C}^*}-Algebra C(K) and the Koopman Operator |pages=45–70 }}

{{Functional analysis}}

{{SpectralTheory}}

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{{DEFAULTSORT:Gelfand-Naimark theorem}}

Category:Operator theory

Category:Theorems in functional analysis

Category:C*-algebras