Gelfand transform

The Gelfand Transform

Let 𝒜 be a Banach algebraMathworldPlanetmath over . Let be the space of all multiplicative linear functionals in 𝒜, endowed with the weak-* topologyMathworldPlanetmath. Let C() denote the algebra of complex valued continuous functionsMathworldPlanetmathPlanetmath in .

The Gelfand transform is the mapping



where x^C() is defined by x^(ϕ):=ϕ(x),ϕ

The Gelfand transform is a continuous homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from 𝒜 to C().

Theorem - Let C0() denote the algebra of complex valued continuous functions in , that vanish at infinity. The image of the Gelfand transform is contained in C0().

The Gelfand transform is a very useful tool in the study of commutative Banach algebras and, particularly, commutativePlanetmathPlanetmathPlanetmath C*-algebras (

Classification of commutative C*-algebras: Gelfand-Naimark theorems

The following results are called the Gelfand-Naimark theoremsMathworldPlanetmath. They classify all commutative C*-algebras and all commutative C*-algebras with identity elementMathworldPlanetmath.

Theorem 1 - Let 𝒜 be a C*-algebra over . Then 𝒜 is *-isomorphic to C0(X) for some locally compact Hausdorff spacePlanetmathPlanetmath X. Moreover, the Gelfand transform is a *-isomorphism between 𝒜 and C0().

Theorem 2 - Let 𝒜 be a unital C*-algebra over . Then 𝒜 is *-isomorphic to C(X) for some compactPlanetmathPlanetmath Hausdorff space X. Moreover, the Gelfand transform is a *-isomorphism between 𝒜 and C().

The above theorems can be substantially improved. In fact, there is an equivalence ( between the category of commutative C*-algebras and the category of locally compact Hausdorff spaces. For more and details about this, see the entry about the general Gelfand-Naimark theorem (

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 C*-algebras
Defines commutative C*-algebras classification
Defines Gelfand-Naimark theorem