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In mathematics, the Gelfand–Naimark theorem states that an arbitrary
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
''A'' is isometrically *-isomorphic to a C*-subalgebra of
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector s ...
s on a Hilbert space. This result was proven by Israel Gelfand and
Mark Naimark Mark Aronovich Naimark (russian: Марк Ароно́вич Наймарк) (5 December 1909 – 30 December 1978) was a Soviet mathematician who made important contributions to functional analysis and mathematical physics. Life Naimark was b ...
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 The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
π''f'' of ''A'' where ''f'' ranges over the set of
pure state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s of A and π''f'' is the irreducible representation associated to ''f'' by the
GNS construction GNS may refer to: Places * Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia * Gainesville station (Georgia), an Amtrak station in Georgia, United States Companies and organizations * Gesellschaft für Nuklear-Service, a German nuclear-wa ...
. Thus the Gelfand–Naimark representation acts on the Hilbert direct sum of the Hilbert spaces ''H''''f'' by : \pi(x) bigoplus_ H_f= \bigoplus_ \pi_f(x)H_f. π(''x'') is a
bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector s ...
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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
s, 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 An operator ''A'' on an (infinite dimensional) Banach space or Hilbert space H has a cyclic vector ''f'' if the vectors ''f'', ''Af'', ''A2f'',... span H. Equivalently, ''f'' is a cyclic vector for ''A'' in case the set of all vectors of the form ' ...
ξ. Since : \begin \, \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 \\ pt& = \langle \xi \mid \pi_f(x^* x) \xi \rangle= f(x^* x) > 0, \end 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 In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
''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\, _ = \sup_f \sqrt 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 \, _
factor Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, su ...
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 In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
''A'' having an approximate identity: : \sup_ f(x^*x) = \sup_ 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 GNS may refer to: Places * Binaka Airport, in Gunung Sitoli, Nias Island, Indonesia * Gainesville station (Georgia), an Amtrak station in Georgia, United States Companies and organizations * Gesellschaft für Nuklear-Service, a German nuclear-wa ...
*
Stinespring factorization theorem In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra ''A'' as a compositi ...
* Gelfand–Raikov theorem * Tannaka–Krein duality


References

* (als
available from Google Books
* , also available in English from North Holland press, see in particular sections 2.6 and 2.7. {{DEFAULTSORT:Gelfand-Naimark theorem Operator theory Theorems in functional analysis C*-algebras