Convex-transitive Douglas algebras

TL;DR

This paper demonstrates that certain Banach algebras, including conformal invariant Douglas algebras, exhibit weak-star convex-transitivity, indicating high symmetry using only inner isometries such as weighted composition operators.

Contribution

It establishes convex-transitivity for specific Banach algebras, highlighting their geometric symmetry and the role of inner isometries like weighted composition operators.

Findings

Conformal invariant Douglas algebras are weak-star convex-transitive.

Symmetry is achieved using only inner isometries.

Weighted composition operators exemplify the symmetry group.

Abstract

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.