I'm wondering why the maximal ideals of the Wiener algebra $\mathcal{W}$ are of the form $\{M_z:z\in \mathbb{T}\}$ where $M_z=\{f\in \mathcal{W}\; |\; f(z)=0\}$.
Given that the Wiener algebra is a commutative Banach algebra we have to find a continuous linear functional $\phi$ on $\mathcal{W}$ such that $\ker(\phi) = M_z$, right? But I have no idea how we do this?
By continuous linear functional in this context we mean a homomorphism of $\mathcal{A}$ into $\mathbb{C}$.