Some conditions on Douglas algebras that imply the invariance of the minimal envelope map

We give general conditions on certain families of Douglas algebras that imply that the minimal envelope of the given algebra is the algebra itself. We also prove that the minimal envelope of the intersection of two Douglas algebras is the intersection of their minimal envelope.

A note on minimal envelopes of Douglas algebras, minimal support sets, and restricted Douglas algebras

We characterize the interpolating Blaschke products of finite type in terms of their support sets. We also give a sufficient condition on the restricted Douglas algebra of a support set that is invariant under the Bourgain map, and its minimal envelope is singly generated.

Douglas algebras without maximal subalgebra and without minimal superalgebra

We give several examples of Douglas algebras that do not have any maximal subalgebra. We find a condition on these algebras that guarantees that some do not have any minimal superalgebra. We also show that ifAis the only maximal subalgebra of a Douglas algebraB, then the algebraAdoes not have any maximal subalgebra.