Gelfand transform
The Gelfand Transform
Let be a Banach algebra over . Let be the space of all multiplicative linear functionals in , endowed with the weak-* topology. Let denote the algebra of complex valued continuous functions in .
The Gelfand transform is the mapping
where is defined by
The Gelfand transform is a continuous homomorphism from to .
Theorem - Let denote the algebra of complex valued continuous functions in , that vanish at infinity. The image of the Gelfand transform is contained in .
The Gelfand transform is a very useful tool in the study of commutative Banach algebras and, particularly, commutative -algebras (http://planetmath.org/CAlgebra).
Classification of commutative -algebras: Gelfand-Naimark theorems
The following results are called the Gelfand-Naimark theorems. They classify all commutative -algebras and all commutative -algebras with identity element.
Theorem 1 - Let be a -algebra over . Then is *-isomorphic to for some locally compact Hausdorff space . Moreover, the Gelfand transform is a *-isomorphism between and .
Theorem 2 - Let be a unital -algebra over . Then is *-isomorphic to for some compact Hausdorff space . Moreover, the Gelfand transform is a *-isomorphism between and .
The above theorems can be substantially improved. In fact, there is an equivalence (http://planetmath.org/EquivalenceOfCategories) between the category of commutative -algebras and the category of locally compact Hausdorff spaces. For more and details about this, see the entry about the general Gelfand-Naimark theorem (http://planetmath.org/GelfandNaimarkTheorem).
Title | Gelfand transform |
---|---|
Canonical name | GelfandTransform |
Date of creation | 2013-03-22 17:22:39 |
Last modified on | 2013-03-22 17:22:39 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 26 |
Author | asteroid (17536) |
Entry type | Definition |
Classification | msc 46L35 |
Classification | msc 46L05 |
Classification | msc 46J40 |
Classification | msc 46J05 |
Classification | msc 46H05 |
Related topic | MultiplicativeLinearFunctional |
Related topic | NoncommutativeTopology |
Related topic | CAlgebra3 |
Related topic | CAlgebra |
Related topic | CompactQuantumGroup |
Defines | classification of commutative -algebras |
Defines | commutative -algebras classification |
Defines | Gelfand-Naimark theorem |