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Gelfand–Naimark theorem

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Mathematics theorem in functional analysis

Not to be confused withGelfond–Schneider theorem.

Inmathematics, theGelfand–Naimark theorem states that an arbitraryC*-algebraA is isometrically *-isomorphic to a C*-subalgebra ofbounded operators on aHilbert space. This result was proven byIsrael Gelfand andMark 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 anoperator algebra.

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The Gelfand–Naimark representation π is the Hilbert space analogue of thedirect sum of representations π_f ofA wheref ranges over the set ofpure states of A and π_f is theirreducible representation associated tof by theGNS construction. Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spacesH_f by

\pi (x)[\bigoplus _{f}H_{f}]=\bigoplus _{f}\pi _{f}(x)H_{f}.

π(x) is abounded 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 isinjective, since for *-morphisms of C*-algebras injective implies isometric. Letx be a non-zero element ofA. By theKrein extension theorem for positivelinear functionals, there is a statef onA such thatf(z) ≥ 0 for all non-negative z inA andf(−x*x) < 0. Consider the GNS representation π_f withcyclic vector ξ. Since

{\begin{aligned}\|\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{aligned}}

it follows that π_f (x) ≠ 0, so π (x) ≠ 0, so π is injective.

The construction of Gelfand–Naimarkrepresentation depends only on the GNS construction and therefore it is meaningful for anyBanach *-algebraA having anapproximate identity. In general (whenA is not a C*-algebra) it will not be afaithful representation. The closure of the image of π(A) will be a C*-algebra of operators called theC*-enveloping algebra ofA. Equivalently, we can define the C*-enveloping algebra as follows: Define a real valued function onA by

\|x\|_{\operatorname {C} ^{*}}=\sup _{f}{\sqrt {f(x^{*}x)}}

asf ranges over pure states ofA. This is a semi-norm, which we refer to as theC* semi-norm ofA. The setI of elements ofA whose semi-norm is 0 forms a two sided-ideal inA closed under involution. Thus thequotient vector spaceA /I is an involutive algebra and the norm

\|\cdot \|_{\operatorname {C} ^{*}}

factors through a norm onA /I, which except for completeness, is a C* norm onA /I (these are sometimes called pre-C*-norms). Taking the completion ofA /I relative to this pre-C*-norm produces a C*-algebraB.

By theKrein–Milman theorem one can show without too much difficulty that forx an element of theBanach *-algebraA 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 onA is to take the above supremum over all states.

The universal construction is also used to defineuniversal C*-algebras of isometries.

Remark. TheGelfand representation orGelfand isomorphism for a commutative C*-algebra with unit $A {\displaystyle A}$ is an isometric *-isomorphism from $A {\displaystyle A}$ 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, ofA with the weak* topology.

References

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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. (alsoavailable from Google Books)
Dixmier, Jacques (1969).Les C*-algèbres et leurs représentations. Gauthier-Villars.ISBN 0-7204-0762-1., 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 ${\operatorname {C} ^{*}}$ -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.

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