Analytic primes, $M$-ideals, and $p$-sets in $H^{\infty }(\mathbb{D})$

Abstract

We investigate the structure of $p$-sets, $M$-ideals, and a newly introduced notion of analytic primes in $H^{\infty }(\mathbb{D})$, where $H^{\infty }(\mathbb{D})$ denotes the Banach algebra of all bounded analytic functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. We prove that $M$-ideals in $H^{\infty }(\mathbb{D})$ are analytic primes and are dense in the Hardy space. Outer functions play a key role in representing closed principal ideals in $H^{\infty }(\mathbb{D})$ that are $M$-ideals. Some of our results apply to the polydisc. The results presented in this paper offer some new perspectives on $H^{\infty }(\mathbb{D})$.

New title and revised version. 33 pages