Gelfand–Naimark theorem

Mathematics theorem in functional analysis

Not to be confused with Gelfond–Schneider theorem.

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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.

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The Gelfand–Naimark representation π is the Hilbert space analogue of 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 _{f}H_{f}]=\bigoplus _{f}\pi _{f}(x)H_{f}.

{\displaystyle \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 π_f with cyclic 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}}

{\displaystyle {\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–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)}}

{\displaystyle \|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} ^{*}}

{\displaystyle \|\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).

{\displaystyle \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 $A$ {\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, of A with the weak* topology.

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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. (also available 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 ${\operatorname {C} ^{*}}$ {\displaystyle {\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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