In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra 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
\( \pi (x)[\bigoplus _{{f}}\xi _{f}]=\bigoplus _{{f}}\pi _{f}(x)\xi _{f}. \)
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
\( {\begin{aligned}\|\pi _{f}(x)\xi \|^{2}&=\langle \pi _{f}(x)\xi \mid \pi _{f}(x)\xi \rangle =\langle \xi \mid \pi _{f}(x^{*})\pi _{f}(x)\xi \rangle \\[6pt]&=\langle \xi \mid \pi _{f}(x^{*}x)\xi \rangle =f(x^{*}x)>0,\end{aligned}} \)
it follows that 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
GNS construction
Stinespring factorization theorem
Gelfand–Raikov theorem
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. (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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
