Interpolating Blaschke products and factorization in Douglas algebras (Q1174977)

In this interesting article the authors prove some important results concerning Douglas algebras of \(\mathbb{T}\). About Douglas algebras see \textit{J. B. Garnett's} book [Bounded analytic functions (1981; Zbl 0469.30024)]. Let \(H^ \infty\) be the Banach algebra of all bounded analytic functions in the open unit disk \(\mathbb{D}=\{z\in\mathbb{C}\): \(| z|<1\}\) and let \(M(H^ \infty)\) denote its maximal ideal space. Let \(P(m)\) denote the Gleason part of \(m\in M(H^ \infty)\). Hoffman showed that any Gleason part \(P(m)\) in \(M(H^ \infty)\) is either a single point or an analytic disk. The set of points in \(M(H^ \infty)\) whose Gleason part reduces to a single point are called trivial points. For Douglas algebras \(B\), the maximal ideal space \(M(B)\) of \(B\) can be viewed as a compact subset of \(M(H^ \infty)\). The authors give a complete solution to the problem of \textit{C. Guillory}, \textit{K. Izuchi} and \textit{D. Sarason} [It was the problem about divisibility structure of Douglas algebras, Proc. R. Ir. Acad., Sect. A 84, 1-7 (1984; Zbl 0559.46022)]. They prove: Let \(u\) be a unimodular function in the Douglas algebra \(B\). Assume that \(u\) does not vanish identically on any nontrivial Gleason part in \(M(B)\). Let \(g\) be a function in \(B\) satisfying one of the following conditions: (a) Every zero of \(u\) is a zero of \(g\) of at least as high multiplicity; (b) \(| g|<| u|\) on \(M(B)\). Then \(g\) is divisible by \(u\) in \(B\). This theorem implies several corollaries. In section 4 the authors present new results about support sets. For example, they prove the following pretty results: a) there exist no minimal support sets containing more than one point; b) the support set of a trivial point \(m\in M(H^ \infty)\setminus M(L^ \infty)\) strictly contains the support set of another trivial point of \(M(H^ \infty) \setminus M(L^ \infty)\). This paper is a big step in the theory of \(M(H^ \infty)\).