Reflexivity of the isometry group of some classical spaces

Abstract

We investigate the reflexivity of the isometry group and the automorphism group of some important metric linear spaces and algebras. The paper consists of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of measurable functions. 4. Hardy spaces. 5. Banach algebras of holomorphic functions. 6. Fréchet algebras of holomorphic functions. 7. Spaces of continuous functions. Introduction.This paper is concerned with the reflexivity of the isometry group and the automorphism group of certain particular, but important, topological vector spaces and algebras. Although we deal mainly with Banach spaces, we are also interested in other (not necessarily locally convex) metric linear spaces and some Fréchet algebras.Reflexivity problems for subalgebras of the algebra of all bounded linear operators acting on a Hilbert space represent one of the most active research areas in operator theory. The study of similar questions concerning sets of linear transformations on Banach algebras rather than on Hilbert spaces was initiated by Kadison [10] and Larson [14]. In [10], motivated by the study of Hochschild cohomology of operator algebras, the reflexivity of the Lie algebra of all derivations on a von Neumann algebra 2000 Mathematics Subject Classification: 47B49, 46B04, 47B10.