= Solution
A <character of an algebra> is a nonzero <multiplicative> complex-<linear map> $\chi:A\to\mathbb C$. A maximal proper ideal is a proper ideal contained in no larger proper ideal. For a character, $\chi(1)=1$ and $\chi(a)\in\sigma(a)$: otherwise $a-\chi(a)1$ would be invertible although its character value is zero. The spectral bound gives $|\chi(a)|\leq\|a\|$, so the character is continuous. It is onto, and its kernel is a closed <maximal ideal> because $A/\ker\chi\cong\mathbb C$.
Conversely let $M$ be maximal. Its closure is an ideal and remains proper: otherwise an element of $M$ would lie within distance one of the unit, hence would be invertible by the <Neumann series>, forcing $1\in M$. Maximality then gives $\overline M=M$. The <Banach algebra> $A/M$ is a field. For any $b$ in this quotient choose $\lambda\in\sigma(b)$, using nonemptiness of the complex spectrum. In a field only zero is noninvertible, so $b-\lambda1=0$. Hence every quotient element is a scalar multiple of the unit. The <quotient map> therefore determines a character $\chi_M$, uniquely normalized by $\chi_M(1)=1$. These constructions are inverse:
$$
\boxed{\Delta(A)\longleftrightarrow M(A),\qquad\chi\longmapsto\ker\chi.}
$$
For the convolution algebra, write $e_j$ for the sequence supported at $j$ with value one. Then $e_j e_k=e_{j+k}$, $e_0=1$, and $e_1e_{-1}=1$. If $z=\chi(e_1)$, continuity and $\|e_{\pm1}\|_1=1$ imply $|z|\leq1$ and $|z^{-1}|\leq1$. Thus $|z|=1$, so uniquely $z=e^{it}$ for $0\leq t<2\pi$, and $\chi(e_j)=e^{ijt}$ for every integer $j$. <Absolute convergence> in $\ell^1$ now gives
$$
\boxed{\chi_t(\alpha)=\sum_{j\in\mathbb Z}\alpha_je^{ijt}.}
$$
Conversely each displayed map is bounded, nonzero, and <multiplicative>, since the absolutely convergent double sum can be rearranged along $k+l=j$. Thus these are all characters, and all maximal proper ideals are
$$
\boxed{M_t=\left\{\alpha\in\ell^1(\mathbb Z):\sum_j\alpha_je^{ijt}=0\right\},\qquad0\leq t<2\pi.}
$$
They are distinct because the character is uniquely recovered from its kernel and its value on the unit.
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