= Character space of an Abelian L1 group algebra
{title2=$\Delta(L^1(G))\cong\widehat G$}
For a locally compact Hausdorff Abelian group, each <continuous unitary character> defines an <algebra character> by $f\mapsto\int f(x)\chi(x)\,dx$. Conversely, if $\Phi$ is a nonzero <algebra character> and $\Phi(g)\ne0$, the ratio $\Phi(T_xg)/\Phi(g)$ is independent of $g$, continuous and multiplicative. Translation <isometries> bound all its integer powers, so its modulus is one. The <convolution> formula $f*g=\int f(x)T_xg\,dx$ recovers $\Phi(f)=\int f\chi$. The <Gelfand topology> corresponds to the <compact-open topology> on the <Pontryagin dual group>; conjugating the characters gives the alternative Fourier sign convention.
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