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Gelfand transform (Definition)

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)$.

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.

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 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.



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See Also: multiplicative linear functional, noncommutative topology, C*-algebras and quantum compact groupoids, $C^*$-algebra, compact quantum group

Also defines:  classification of commutative $C^*$-algebras, commutative $C^*$-algebras classification, Gelfand-Naimark theorem
Keywords:  Gelfand transform, C*-algebra, C*- algebra representations, CompactQuantumGroup
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Cross-references: category, Hausdorff space, compact, unital, locally compact Hausdorff space, identity element, commutative, contained, image, vanish at infinity, homomorphism, mapping, continuous functions, complex, algebra, weak-* topology, multiplicative linear functionals, Banach algebra
There are 5 references to this entry.

This is version 23 of Gelfand transform, born on 2007-07-05, modified 2008-11-08.
Object id is 9743, canonical name is GelfandTransform.
Accessed 1522 times total.

Classification:
AMS MSC: 46H05 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: General theory of topological algebras)
 46J05 (Functional analysis :: Commutative Banach algebras and commutative topological algebras :: General theory of commutative topological algebras)
 46J40 (Functional analysis :: Commutative Banach algebras and commutative topological algebras :: Structure, classification of commutative topological algebras)
 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
 46L35 (Functional analysis :: Selfadjoint operator algebras :: Classifications of $C^*$-algebras, factors)
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