# Gelfand transform

## The Gelfand Transform

Let $\mathcal{A}$ be a Banach algebra^{} over $\u2102$.
Let $\u25b3$ be the space of all
multiplicative linear functionals in $\mathcal{A}$, endowed with the weak-* topology^{}. Let
$C(\u25b3)$ denote the algebra of complex valued continuous functions^{} in $\u25b3$.

where $\widehat{x}\in C(\u25b3)$ is defined by $\widehat{x}(\varphi ):=\varphi (x),\forall \varphi \in \u25b3$

The Gelfand transform is a continuous homomorphism^{} from $\mathcal{A}$ to $C(\u25b3)$.

Theorem - Let ${C}_{0}(\u25b3)$ denote the algebra of complex valued continuous functions in $\u25b3$, that vanish at infinity. The image of the Gelfand transform is contained in ${C}_{0}(\u25b3)$.

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 $\u2102$. 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 ${C}_{0}(\u25b3)$.

Theorem 2 - Let $\mathcal{A}$ be a unital ${C}^{*}$-algebra over $\u2102$. 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 $C(\u25b3)$.

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 |