Division theorems and the Shilov property for \(H^\infty+C\) (Q1805969)

Let \(T\) be the boundary of the unit disk \(\mathbb{D}= \{z\in\mathbb{C}: |z|\leq 1\}\), \(C\) the space of continuous, complex valued functions on \(T\) and \(H^\infty\) the uniform algebra of boundary values of bounded analytic functions in \(\mathbb{D}\). Spectra of uniform algebras \(A\) are denoted by \(M(A)\). In this paper we present some results on the ideal structure of the algebra \(H^\infty +C\), and give some applications to divisibility problems in this space. The main result shows that \(H^\infty+C\) has the Shilov property, meaning that any ideal \(I\) in \(H^\infty+C\) contains every function vanishing identically in a neighhorhood of the hull \(Z(I)=\{m\in M(H^\infty+C):f(m)=0\) for every \(f\in I\}\) of \(I\) -- a property shared by every regular algebra. As a corollary, one sees that a function \(f\) in \(H^\infty+C\) vanishing in a neighborhood of the zeros (in \(M(H^\infty+C))\), of another function \(g\in H^\infty+C\) is divisible by \(g\). Previous results on divisibility and factorization in \(H^\infty +C\), or more generally, in Douglas algebras, have been obtained e.g. by \textit{S. Axler}, \textit{P. Gorkin} [Mich. Math. J. 31, 89-94 (1994; Zbl 0597.46054)], \textit{C. Guillory}, \textit{K. Izuchi}, \textit{D. Sarason} [Proc. R. Ir. Acad., Sect. A 84, 1-7 (1984; Zbl 0559.46022)], \textit{C. Guillory} and \textit{D. Sarason} [Mich. Math. J. 28, 173-181 (1981; Zbl 0456.30031), \textit{K. Izuchi}, \textit{Y. Izuchi} [Mich. Math. J. 33, 435-443 (1986; Zbl 0614.46045)] and \textit{P. Gorkin}, \textit{R. Mortini} [Mich. Math. J. 38, No. 1, 147-160 (1991; Zbl 0781.46037)]. The paper concludes with several interesting open questions on the local structure of the algebra \(H^\infty+C\).