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

Every character of an algebra on ${\ell }^1(\mathbb{Z})$ with convolution multiplication is uniquely of this form for $0\le t<2\pi$. If $e_j$ is the point mass at $j$, then $e_j{e}_k=e_{j+k}$. A character has norm one and $e_1{e}_{-1}=1$, so $|\chi (e_1)|\le 1$ and $|\chi (e_1{)}^{-1}|\le 1$. Hence $\chi (e_1)$ lies on the unit circle; multiplication determines all $\chi (e_j)$ and continuity determines the displayed series. Conversely absolute convergence permits rearrangement of the double series, proving multiplicativity. The maximal ideals are exactly their kernels: each consists of sequences whose Fourier series vanishes at one fixed point of the unit circle.

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.