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.