I'm working on an exercise (8.31) in the book Smooth Manifolds and Observables by Jet Nestruev. Let me reformulate the exercise as follows:
Let $\mathcal F$ be the algebra of bounded holomorphic functions on the unit open disk $B(0,1)$. Show that the maximal ideals of $\mathcal F$ are exactly one of the following two types:
- $\mathfrak{m}_x$, where $|x|<1$. This is the ideal of functions in $\mathcal F$ vanishing at $x$.
- $\mathfrak{m}_x$, where $|x|=1$. This is the ideal of functions in $\mathcal F$ whose limit as $z\to x$ exists and equals $0$.
First I'm having trouble showing that $\mathfrak{m}_x$ is a maximal ideal when $|x|=1$. This amounts to proving that if $f$ is bounded and holomorphic on $B(0,1)$ with $\lim_{z\to x}f(z)$ unequal to $0$ (possibly does not exist), then there is $g\in\mathcal F$ such that the limit of $fg$ at $z=x$ is $1$.
I also don't know how to prove that these are all possible maximal ideals in $\mathcal F$. Any help (hint) would be appreciated.