M-ideals in H^∞(D)

This article intends to initiate an investigation into the structure of M-ideals in H^∞(D), where H^∞(D) denotes the Banach algebra of all bounded analytic functions on the open unit disc D in C. We introduce the notion of analytic primes and prove that M-ideals in H^∞(D) are analytic primes. From function Hilbert space perspective, we additionally prove that M-ideals in H^∞(D) are dense in the Hardy space. We show that outer functions play a key role in representing singly generated closed ideals in H^∞(D) that are M-ideals. This is also relevant to M-ideals in H^∞(D) that are finitely generated closed ideals in H^∞(D). We analyze p-sets of H^∞(D) and their connection to the {S}ilov boundary of the maximal ideal space of H^∞(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.