Character space of continuous functions on the circle (${\Phi }_{C(\mathbb{T})}\cong \mathbb{T}$)

A maximal ideal $M$ of $C(\mathbb{T})$ has a common zero: otherwise finitely many $f_j\in M$ have no simultaneous zero, and ${\sum }_j|f_j{|}^2\in M$ is invertible, a contradiction. If the common zero is $z$, then $M$ is the kernel of evaluation at $z$ by maximality. Thus every algebra character is point evaluation. Continuous functions separate distinct points of the unit circle, and the evaluation bijection is a homeomorphism for the Gelfand topology.