L1 convolution algebra ($(L^1(G),\ast )$)

With left Haar measure, $(f\ast g)(x)={\int }_{G}f(y)g(y^{-1}x)dm(y)$ makes $L^1(G)$ a Banach algebra, since $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$. For an Abelian group it is commutative, and the involution is $f^{\ast }(x)=\overline{f(x^{-1})}$. Its nonzero characters of an algebra are exactly integration against inverses of continuous unitary group characters.

// Target: geometry-and-topology.bigb

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 $\Phi (g)\ne 0$, the ratio $\Phi (T_{x}g)/\Phi (g)$ is independent of $g$, continuous and multiplicative. Translation isometries bound all its integer powers, so its modulus is one. The convolution formula $f\ast g=\int f(x)T_{x}gdx$ 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.