|
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 $C(\bigtriangleup)$ .
Theorem - 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.
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 between the category of commutative $C^*$ -algebras and the category of locally compact Hausdorff spaces. For more information and details about this, see the entry about the general Gelfand-Naimark theorem.
| 
