# Gelfand transform

## The Gelfand Transform

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

The Gelfand transform is the mapping

$\widehat{}\;\;:\mathcal{A}\longrightarrow C(\bigtriangleup)$

$x\longmapsto\widehat{x}$

where $\widehat{x}\in C(\bigtriangleup)$ is defined by $\;\;\widehat{x}(\phi):=\phi(x),\;\;\;\forall\phi\in\bigtriangleup$

The Gelfand transform is a continuous homomorphism from $\mathcal{A}$ to $C(\bigtriangleup)$.

Let $C_{0}(\bigtriangleup)$ denote the algebra of complex valued continuous functions in $\bigtriangleup$, that vanish at infinity. The image of the Gelfand transform is contained in $C_{0}(\bigtriangleup)$.

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 $\mathcal{A}$ be a $C^{*}$-algebra over $\mathbb{C}$. Then $\mathcal{A}$ is *-isomorphic to $C_{0}(X)$ for some locally compact Hausdorff space $X$. Moreover, the Gelfand transform is a *-isomorphism between $\mathcal{A}$ and $C_{0}(\bigtriangleup)$.

Theorem 2 - Let $\mathcal{A}$ be a unital $C^{*}$-algebra over $\mathbb{C}$. Then $\mathcal{A}$ is *-isomorphic to $C(X)$ for some compact Hausdorff space $X$. Moreover, the Gelfand transform is a *-isomorphism between $\mathcal{A}$ and $C(\bigtriangleup)$.

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 GelfandTransform 2013-03-22 17:22:39 2013-03-22 17:22:39 asteroid (17536) asteroid (17536) 26 asteroid (17536) Definition msc 46L35 msc 46L05 msc 46J40 msc 46J05 msc 46H05 MultiplicativeLinearFunctional NoncommutativeTopology CAlgebra3 CAlgebra CompactQuantumGroup classification of commutative $C^{*}$-algebras commutative $C^{*}$-algebras classification Gelfand-Naimark theorem