Wiener algebra ($A(\mathbb{T})$)

The Wiener algebra consists of periodic functions with absolutely summable Fourier coefficients, with norm ${\sum }_k|a_k|$. Multiplication corresponds to discrete convolution of coefficients, so the norm is submultiplicative. Consequently $e^f={\sum }_{n\ge 0}{f}^n/n!$ also belongs to the algebra when $f$ does. Every such Fourier series is uniformly convergent and defines a continuous function.

Every algebra character of the Wiener algebra is evaluation at a unique point of the unit circle. Writing $u(w)=w$, both $u$ and $u^{-1}$ have norm one. Thus $z=\varphi (u)$ satisfies $|z|\le 1$ and $|z^{-1}|\le 1$, hence $|z|=1$. Continuity then gives $\varphi ({\sum }_{n\in \mathbb{Z}}{a}_n{u}^n)=\sum a_n{z}^n$. Conversely, evaluation defines a continuous algebra character. The resulting bijection is a homeomorphism for the Gelfand topology.

The analytic Wiener algebra consists of the power series $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$ with ${\sum }_{n\ge 0}|a_n|<\infty$, identified with their values on the unit circle. The norm is $\Vert f\Vert =\sum |a_n|$. Discrete convolution gives multiplication and a submultiplicative norm; completeness follows from the absolutely summable sequence space. Each series converges uniformly on the closed unit disk and is holomorphic in its interior.

Every algebra character of the analytic Wiener algebra is evaluation at a unique point $z$ of the closed unit disk. Indeed, for $u(w)=w$, continuity and multiplicativity give $z=\varphi (u)$, $|z|\le \Vert u\Vert =1$ and $\varphi (f)={\sum }_{n\ge 0}{a}_n{z}^n$. Conversely, this formula defines a continuous algebra character for every such $z$. The evaluation correspondence is a homeomorphism for the Gelfand topology, since the power series are uniformly convergent on the closed unit disk.