Sign up ×  Let $G = \mathbb Z^n$, the fi nite cyclic group of
order $n$, and let $A = l ^1(G)$, a Banach algebra over $\mathbb
C$ when the product is convolution, de fined for $f, g \in A$ by
$f * g(x) =\sum _{y \in G}f(y)g(x -y)$. Can we determine the
maximal ideal space of $A$, equivalently the space of algebra
homomorphisms from $A$ onto $\mathbb C$?  share|cite|improve
this question Is there something missing or unclear in the
answer below? –  Martin Feb 15 '13 at 9:25  1 Answer 1  up vote
5 down vote accepted  The group $G$ embeds into $\ell^1(G)$ via
the homomorphism $g \mapsto \delta_g$ since $\delta_g \ast
\delta_h = \delta_{gh}$. Since $\lVert \delta_g \rVert_1 = 1$, a
surjective homomorphism $\ell^1(G) \to \mathbb{C}$ restricts to
a homomorphism $G \to \mathbb{C}^\times$ with bounded range. The
maximal bounded subgroup of $\mathbb{C}^\times$ is $\mathbb{T} =
\{z \in \mathbb{C} \mid |z| =1\}$, so we obtain a map from the
maximal ideal space of $\ell^1(G)$ to the Pontryagin dual
$\widehat{G} =\operat