Convex-transitive Douglas algebras

The convex-transitivity property can be seen as a convex generalization of the almost transitive (or quasi-isotropic) group action of the isometry group of a Banach space on its unit sphere. We will show that certain Banach algebras, including conformal invariant Douglas algebras, are weak-star convex-transitive. Geometrically speaking, this means that the investigated spaces are highly symmetric. Moreover, it turns out that the symmetry property is satisfied by using only `inner' isometries, i.e. a subgroup consisting of isometries which are homomorphisms on the algebra. In fact, weighted composition operators arising from function theory on the unit disk will do. Some interesting examples are provided at the end.