Let $W$ be the space of all complex functions on $[0,2\pi[$ whose Fourier series converges absolutely, that is, all functions $f:[0,2\pi[\longrightarrow\mathbb{C}$ whose Fourier series

is such that $\sum_{{n}}|\hat{f}(n)|<\infty$ .

Under pointwise operations and the norm $\|f\|=\sum_{{n}}|\hat{f}(n)|\,,$ $W$ is a commutative Banach algebra of continuous functions, with an identity element. $W$ is usually called the Wiener algebra.

Theorem - $W$ is isometrically isomorphic to the Banach algebra $\ell^{1}(\mathbb{Z})$ with the convolution product. The isomorphism is given by:

Theorem (Wiener) - If $f\in W$ has no zeros then $1/f\in W$, that is, $1/f$ has an absolutely convergent Fourier series.

Proof : We want to prove that $f$ is invertible in $W$. As $W$ is commutative, that is the same as proving that $f$ does not belong to any maximal ideal of $W$. Therefore we only need to show that $f$ is not in the kernel of any multiplicative linear functional of $W$.

Let $\phi$ be a multiplicative linear functional in $W$. We have that

Since $\|\phi\|=1$ we have that

and

Since $1=|\phi(e^{{it}}e^{{-it}})|=|\phi(e^{{it}})||\phi(e^{{-it}})|$ we deduce that

We can conclude that

$\phi(e^{{it}})=e^{{it_{0}}}\;$ for some $t_{0}\in[0,2\pi[$

Therefore we obtain

which is non-zero by definition of $f$.

We conclude that $f$ does not belong to the kernel of any multiplicative linear functional $\phi$. $\square$

The Wiener algebra is a Banach *-algebra with the involution given by $f^{*}(t):=\overline{f(-t)}$, but it is not a $C^{*}$-algebra under this involution.