= Character space of the Wiener algebra
{title2=$\Phi_{A(\mathbb T)}\cong\mathbb T$}
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 $z=\phi(u)$ satisfies $|z|\leq1$ and $|z^{-1}|\leq1$, hence $|z|=1$. Continuity then gives $\phi(\sum_{n\in\mathbb Z}a_nu^n)=\sum a_nz^n$. Conversely, evaluation defines a continuous <algebra character>. The resulting bijection is a <homeomorphism> for the <Gelfand topology>.
Back to article page
