What is the maximal ideal space of $H^\infty$?

Question

What is the spectrum of $H^\infty$, the Banach algebra of all bounded holomorphic functions in the open unit disk $D=\{z\in \mathbb{C}\mid |z| <1 \} $?

Answer 1

Edit: This is an answer to the previous version of this question.

This algebra is complete; it is a simple application of Morera's theorem which you may use to show that the uniform limit of such functions is actually holomorphic. This algebra is traditionally denoted by $H^\infty$ and is highly non-separable.

For this reason, the maximal ideal space of $H^\infty$ is huge. See this article.

Answer 2

There's no simple description of the spectrum of the algebra of bounded holomorphic functions in the disk (known as $H^\infty$). It's an Axiom-of-Choice-ish thing.

If $|z|<1$ then $f\mapsto f(z)$ is a complex homomorphism, so the open disk is contained in the spectrum in a natural way. The Corona Theorem says that the disk is dense. This is one of the huge theorems that made Lennart Carleson Lennart Carleson. Carleson's proof is very nasty, but the techniques led to a lot of stuff about the unit disk. See Garnett Bounded Analytic Functions. Somewhat later Tom Wolfe gave a much simpler proof.