Douglas algebras without maximal subalgebra and without minimal superalgebra

Abstract

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 if $A$ is the only maximal subalgebra of a Douglas algebra $B$, then the algebra $A$ does not have any maximal subalgebra.