Chapter 16 Uniform algebras as Banach spaces

Abstract

Any Banach space can be realized as a direct summand of a uniform algebra, and one does not expect an arbitrary uniform algebra to have an abundance of properties not common to all Banach spaces. One general result concerning arbitrary uniform algebras is that no proper uniform algebra is linearly homeomorphic to a C(K) -space. Nevertheless many specific uniform algebras arising in complex analysis share (or are suspected to share) certain Banach space properties of C(K) . We discuss the family of tight algebras, which includes algebras of analytic functions on strictly pseudoconvex domains and algebras associated with rational approximation theory in the plane. Tight algebras are in some sense close to C(K) -spaces, and along with C(K) -spaces they have the Pelczynski and the Dunford-Pettis properties. We also focus on certain properties of C(K) -spaces that are inherited by the disk algebra. This includes a discussion of interpolation between H p -spaces and Bourgain's extension of Grothendieck's theorem to the disk algebra. We conclude with a brief description of linear deformations of uniform algebras and a brief survey of the known classification results.