Editing 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 operator]]s 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]] π<sub>''f''</sub> of ''A'' where ''f'' ranges over the set of [[State (functional analysis)#Pure states|pure states]] of A and π<sub>''f''</sub> 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''<sub>''f''</sub> by

:<math> \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 functional]]s, there is a state ''f'' on ''A'' such that ''f''(''z'') ≥ 0 for all non-negative z in ''A'' and ''f''(&minus;''x''* ''x'') < 0.  Consider the GNS representation π<sub>''f''</sub> with [[cyclic vector]] ξ. Since

:<math>
\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}
</math>

it follows that π<sub>''f''</sub> (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> \|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 space (linear algebra)|quotient vector space]] ''A'' / '''I''' is  an involutive algebra and the norm

:<math> \| \cdot \|_{\operatorname{C}^*} </math>

[[factorization|factor]]s 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> \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*-algebra]]s of isometries.

'''Remark'''. The [[Gelfand representation]] or  [[Gelfand isomorphism]] for a commutative C*-algebra with unit <math>A</math> is an isometric *-isomorphism from <math>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 ==
{{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 <math>{\operatorname{C}^*}</math>-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]]