$M$-ideals in $H^{\mathrm{\infty }}(\mathbb{D})$

This article intends to initiate an investigation into the structure of $M$-ideals in $H^{\mathrm{\infty }}(\mathbb{D})$, where $H^{\mathrm{\infty }}(\mathbb{D})$ denotes the Banach algebra of all bounded analytic functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. We introduce the notion of analytic primes and prove that $M$-ideals in $H^{\mathrm{\infty }}(\mathbb{D})$ are analytic primes. From Hilbert function space perspective, we additionally prove that $M$-ideals in $H^{\mathrm{\infty }}(\mathbb{D})$ are dense in the Hardy space. We show that outer functions play a key role in representing singly generated closed ideals in $H^{\mathrm{\infty }}(\mathbb{D})$ that are $M$-ideals. This is also relevant to $M$-ideals in $H^{\mathrm{\infty }}(\mathbb{D})$ that are finitely generated closed ideals in $H^{\mathrm{\infty }}(\mathbb{D})$. We analyze $p$-sets of $H^{\mathrm{\infty }}(\mathbb{D})$ and their connection to the Šilov boundary of the maximal ideal space of $H^{\mathrm{\infty }}(\mathbb{D})$. Some of our results apply to the polydisc. In addition to addressing questions regarding $M$-ideals, the results presented in this paper offer some new perspectives on bounded analytic functions.