# dual group of $G$ is homeomorphic to the character space of ${L}^{1}(G)$

Let $G$ be a locally compact abelian^{} (http://planetmath.org/AbelianGroup2) group (http://planetmath.org/TopologicalGroup) and ${L}^{1}(G)$ its group algebra^{}.

Let $\widehat{G}$ denote the Pontryagin dual of $G$ and $\mathrm{\Delta}$ the character space of ${L}^{1}(G)$, i.e. the set of multiplicative linear functionals of ${L}^{1}(G)$ endowed with the weak-* topology^{}.

Theorem - The spaces $\widehat{G}$ and $\mathrm{\Delta}$ are homeomorphic. The homeomorphism is given by

$$\omega \u27fc{\varphi}_{\omega},\omega \in \widehat{G}$$ |

where ${\varphi}_{\omega}\in \mathrm{\Delta}$ is defined by

$${\varphi}_{\omega}(f):={\int}_{G}f(s)\omega (s)\mathit{d}\mu (s),f\in {L}^{1}(G)$$ |

Title | dual group of $G$ is homeomorphic to the character space of ${L}^{1}(G)$ |
---|---|

Canonical name | DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G |

Date of creation | 2013-03-22 17:42:49 |

Last modified on | 2013-03-22 17:42:49 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46K99 |

Classification | msc 43A40 |

Classification | msc 43A20 |

Classification | msc 22D20 |

Classification | msc 22D15 |

Classification | msc 22D35 |

Classification | msc 22B10 |

Classification | msc 22B05 |

Related topic | L1GIsABanachAlgebra |