Division in Douglas algebras and some applications
Let A and B be two Douglas algebras and let \(h\in B\). It is proved that the following are equivalent: (1) \(hA\subset B;\) (2) \(z_ B(h)\supset M(B)\setminus M(A)\), here \(z_ B(h)=\{x\in M(B):\) \(h(x)=0\}\) and M(B) is the maximal indeal space of B. If b is an inner function, we put \(N_ B(\bar b)=the\quad closure\quad \cup \{Supp \mu_ x:\quad x\in M(B),\quad | b(x)| <1\}.\) If \(B\subsetneqq A\), we put \(N_ B(A)=\) The closure \(\cup\{Supp \mu_ x: x\in M(B)\setminus M(A)\}.\) The author showed that if \(B\subsetneqq A\), B admits a best approximation and there exists an inner function b such that \(\bar b\in A\) and \(N_ B(A)=N_ B(\bar b)\) then the unit ball of (A/B) has extreme points. In particular, if b is a non continuous inner function then the unit ball of \(H^{\infty}[\bar b]/H^{\infty}+C\) has extreme points.
scientific article; zbMATH DE number 3903008