The first non-trivial example of a uniform algebra which comes to mind is the disc algebra $A(\mathbb{D})$. In a similar manner one can define its relatives $P(U)$ and $R(U)$, where $U$ is any region on the complex plane and $P$, $R$ stand for the closures, of respectively polynomials and rational functions on $U$.

What if we go beyond the case when spectrum is contained in the complex plane? Rudin proved, that if $K$ is scattered, then there are no non-trivial uniform subalgebras of $C(K)$.

Are there any *classical* or *canonical* examples of uniform algebras with non-metrizable spectra or is the theory of uniform algebras just a daughter of complex analysis?

Forgive me if this question seems to be to vague.

Addendum: A uniform algebra is a closed subalgebra of a commutative C*-algebra $C(K)$ which contains constant functions and separates points in $K$.