Gelfand–Naimark theorem

The Gelfand - Neumark -phrases (after Israel Gelfand and Mark Neumark ) and the GNS - construction are the cornerstones of the mathematical theory of C * - algebras. Connect abstractly defined C *-algebras with concrete algebras of functions and operators.

The first examples of C * - algebras that can be specified directly after the definition, the algebra of continuous functions on a locally compact Hausdorff space X which vanish at infinity (see C0 - function), and the sub -C * algebras of, the algebra of bounded linear operators on a Hilbert space H.

Show The Gelfand - Neumark phrases that this is already all possible C *-algebras up to isometric * isomorphism. These results are surprising, since in the definition of C *-algebras is not locally compact Hausdorff spaces or Hilbert spaces of the speech.

• 2.1 Construction of the Hilbert space
• 2.2 Remarks
• 2.3 set of Segal
• 2.4 Further Remarks

Gelfand - Neumark, commutative case

If A is a commutative C * - algebra, then there is a locally compact Hausdorff space X and an isometric * - isomorphism between A and.

Construction of the locally compact Hausdorff space

X is the amount of all of the zero * Figure homomorphisms. For each defined by a figure. Finally, one can prove that the topology of pointwise convergence X makes it a locally compact Hausdorff space and that an isometric * - isomorphism between A and.

Comments

After this block, an element of a commutative C * - algebra can be treated as a continuous function, which can be extended to the so-called continuous functional calculus. Thus, for example, the spectrum of an element is nothing more than the completion of the image of the corresponding continuous function.

This sentence opens up a very fruitful interaction between algebraic properties of C * - algebras and topological properties of locally compact spaces. Is, then one has, among many others following correspondences:

• A has a unit element. X is compact.
• A is finitely generated. X is homeomorphic to a subset of a finite dimensional vector space.
• A is separable. X satisfies the second axiom of countability.
• A has a countable approximation of the One X is σ - compact.
• The adjunction of the element corresponds to the Einpunktkompaktifizierung of X.
• The transition to the multipliers algebra corresponds to the Stone - Čech compactification.

Topological conceptions are translated into algebraic properties of *-algebras commutative C, and then generalized to non-commutative C *-algebras; which is often the starting point for further theories. For this reason is called the theory of C *-algebras as non-commutative topology.

Gelfand - Neumark, general case

If A is a C * - algebra, then there is a Hilbert space H, such that A * is isometric isomorphic to a sub -C *-algebra of L ( H).

Construction of the Hilbert space

Be a continuous linear functional with and for all. Such functionals are also called states of A. For state laws. Then the formula defined an inner product on the quotient space. The completion with respect to this scalar product is a Hilbert space. For each allows the mapping to a continuous linear operator on continue. Then one shows that the so- stated figure is a * - homomorphism. Finally, one constructed from the whole of the Hilbert spaces obtained in a Hilbert space of desired type

Comments

An element of an abstractly defined C *-algebra can thus be treated on a Hilbert space as a bounded linear operator.

The construction of f from above is called the GNS construction, with GNS stands for Gelfand, Neumark and Segal.

It is called * - homomorphisms of the type, representations of A on H. According to the above theorem, each C * - algebra has a faithful (ie injective ) representation on a Hilbert space. A representation is called topologically irreducible if there is no real different from 0 closed subspace U of H for which applies to all.

Set of Segal

If A is a C * - algebra, then the state space S (A) is convex and is a extreme point if and only if the representation is topologically irreducible.

Every irreducible representation of A is of the form for extremal state f of A.

Further remarks

On this basis, a very far-reaching representation theory for C *-algebras has been developed. C *-algebras can be classified by the images of their irreducible representations further. So called a C *-algebra liminal, when the image of each irreducible representation of the algebra of compact operators coincide. A C * - algebra is called postliminal if the image of each irreducible representation contains the algebra of compact operators.