Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 923-937

CONVEX-TRANSITIVE DOUGLAS ALGEBRAS

Maria J. Martin and Jarno Talponen

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, FI-80101 Joensuu, Finland; maria.martin 'at' uef.fi

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, FI-80101 Joensuu, Finland; talponen 'at' iki.fi

Abstract. he 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.

2010 Mathematics Subject Classification: Primary 47B33, 46J15, 30H50, 30J05.

Key words: Banach algebras, isometry group, Douglas algebras, convex-transitive, Hardy space, rotation problem, Nevanlinna class, conformal invariance.

Reference to this article: M.J. Martin and J. Talponen: Convex-transitive Douglas algebras. Ann. Acad. Sci. Fenn. Math. 40 (2015), 923-937.

Full document as PDF file

doi:10.5186/aasfm.2015.4050