Gelfand transform

The Gelfand Transform

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

The Gelfand transform is the mapping

$\widehat{}:𝒜⟶C(△)$

$x⟼\widehat{x}$

where $\widehat{x}\in C(△)$ is defined by $\widehat{x}(\varphi ):=\varphi (x),\forall \varphi \in △$

The Gelfand transform is a continuous homomorphism from $𝒜$ to $C(△)$.

Theorem - Let $C_0(△)$ denote the algebra of complex valued continuous functions in $△$, that vanish at infinity. The image of the Gelfand transform is contained in $C_0(△)$.

The Gelfand transform is a very useful tool in the study of commutative Banach algebras and, particularly, commutative $C^{*}$-algebras (http://planetmath.org/CAlgebra).

Classification of commutative $C^{*}$-algebras: Gelfand-Naimark theorems

The following results are called the Gelfand-Naimark theorems. They classify all commutative $C^{*}$-algebras and all commutative $C^{*}$-algebras with identity element.

Theorem 1 - Let $𝒜$ be a $C^{*}$-algebra over $ℂ$. Then $𝒜$ is *-isomorphic to $C_0(X)$ for some locally compact Hausdorff space $X$. Moreover, the Gelfand transform is a *-isomorphism between $𝒜$ and $C_0(△)$.

Theorem 2 - Let $𝒜$ be a unital $C^{*}$-algebra over $ℂ$. Then $𝒜$ is *-isomorphic to $C(X)$ for some compact 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 (http://planetmath.org/EquivalenceOfCategories) 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 (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 $C^{*}$-algebras
Defines	commutative $C^{*}$-algebras classification
Defines	Gelfand-Naimark theorem