# 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 $\hat{G}$ denote the Pontryagin dual of $G$ and $\Delta$ 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 $\hat{G}$ and $\Delta$ are homeomorphic. The homeomorphism is given by

 $\omega\longmapsto\phi_{\omega}\;,\qquad\qquad\omega\in\hat{G}$

where $\phi_{\omega}\in\Delta$ is defined by

 $\phi_{\omega}(f):=\int_{G}f(s)\omega(s)\;d\mu(s)\;,\qquad\qquad f\in L^{1}(G)$
Title dual group of $G$ is homeomorphic to the character space of $L^{1}(G)$ DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G 2013-03-22 17:42:49 2013-03-22 17:42:49 asteroid (17536) asteroid (17536) 5 asteroid (17536) Theorem msc 46K99 msc 43A40 msc 43A20 msc 22D20 msc 22D15 msc 22D35 msc 22B10 msc 22B05 L1GIsABanachAlgebra