Gelfand transform

The Gelfand Transform

Let $𝒜$ be a Banach algebra over $ℂ$.Let $△$ be the space of allmultiplicative 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).