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

$$\omega ⟼{\varphi }_{\omega },\omega \in \widehat{G}$$

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

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

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