Relation between the maximal ideals of $A(\mathbb{D})$ and $H^{\infty}(\mathbb{D})$

Question

We know that the disk algebra $A(\mathbb{D})$ is a closed subalgebra of $H^{\infty}(\mathbb{D})$. My questions is if there is any relation between:

a) Maximal ideals of those spaces.

b) Multiplicative linear functionals of those spaces.

Any suggestion or reference will be helpful.

Thanks.