Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra

Question

Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$ that it inherits as the dual of $L^{1}(\mathbb{T}).$ Under this topology $H^{\infty}(\mathbb{D})$ is a weak* closed subspace of $L^{\infty}(\mathbb{T}).$ I have the following question:

Does there exist a subspace $M\subseteq H^{\infty}(\mathbb{D})$ which is not weak* closed and contains a nontrivial weak* closed unital subalgebra of $H^{\infty}(\mathbb{D})?$

By nontrivial I mean it contains a nonconstant holomorphic function.

Answer 1

I discussed with some of my friends and found this example. Let $\mathcal{P}$ denotes the set of all polynomials, and let $\mathcal{U}:=\{\varphi(z^2):\varphi\in H^{\infty}(\mathbb{D})\}.$ Set $$M=\mathcal{P}+\mathcal{U}.$$ Note that $M$ contains $\mathcal{U}$ which is weak* closed. Then as the weak* closure of polynomials is dense in $H^{\infty}(\mathbb{D}),$ the weak* closure of $M$ is $H^{\infty}(\mathbb{D}).$ But $M$ is not $H^{\infty}(\mathbb{D}).$ To see that note $\sin{z}\in H^{\infty}(\mathbb{D})$ but $\sin{z}$ can not be written as a sum of a polynomial and an even function in $H^{\infty}(\mathbb{D}).$