  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.
