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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 contin ...
''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 on a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
. 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 In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study o ...
.


Details

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 In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
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_ 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 ...
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 In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
, 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 mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
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 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 ''A'' having an approximate identity. In general (when ''A'' is not a C*-algebra) it will not be a
faithful representation In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group (mathematics), group on a vector space is a linear representation in which different elements of are represented by ...
. 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 \, _ 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 In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitra ...
one can show without too much difficulty that for ''x'' an element of the Banach *-algebra ''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 In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-al ...
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 The Gel'fand–Raikov (Гельфанд–Райков) theorem is a theorem in the mathematics of locally compact topological groups. It states that a locally compact group is completely determined by its (possibly infinite dimensional) unitary ...
* Koopman operator * 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