Gel'fand-Naimark theorem

From Academic Kids

(Redirected from Gelfand-Naimark theorem)

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 a significant point in the development of the theory of C*-algebras in the early 1940s since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an algebra of operators.

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> \pi(x) [\bigoplus_{f} \xi_f] = \bigoplus_{f} \pi_f(x)\xi_f.<math>

Note that 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> \|\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 = \langle \xi \mid \pi_f(x^* x) \xi \rangle= f(x^* x) > 0, <math>

it follows that πf ≠ 0. Injectivity of π follows.

The construction of Gelfand-Naimark representation depends only on the GNS construction and therefore it is meaningful for any B*-algebra A having an approximate identity. In general 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> \|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> || \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 B*-algebra A having an approximate identity:

<math> \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>A<math> is an isometric *-isomorphism from <math>A<math> to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals of A with the weak* topology.

Reference

  • I. M. Gelfand and M. A. Naimark, On the imbeding of normed rings into the ring of operators on a Hilbert space, Math. Sbornik, vol 12, 1943.
Navigation

Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools