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 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
- [math]\displaystyle{ \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 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
- [math]\displaystyle{ \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) \gt 0, \end{align} }[/math]
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
- [math]\displaystyle{ \|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 vector space A / I is an involutive algebra and the norm
- [math]\displaystyle{ \| \cdot \|_{\operatorname{C}^*} }[/math]
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:
- [math]\displaystyle{ \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*-algebras of isometries.
Remark. The Gelfand representation or Gelfand isomorphism for a commutative C*-algebra with unit [math]\displaystyle{ A }[/math] is an isometric *-isomorphism from [math]\displaystyle{ 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
- 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. http://mi.mathnet.ru/eng/msb6155. (also available from Google Books)
- Dixmier, Jacques (1969). Les C*-algèbres et leurs représentations. Gauthier-Villars. ISBN 0-7204-0762-1. https://archive.org/details/calgebras0000dixm., 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 [math]\displaystyle{ {\operatorname{C}^*} }[/math]-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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