Boundaries for spaces of holomorphic functions on M-ideals in their biduals

Abstract

For a complex Banach space X, let A u (B X ) be the Banach algebra of all complex valued functions defined on B X that are uniformly continuous on B X and holomorphic on the interior of B X , and let A wu (B X ) be the Banach subalgebra consisting of those functions in A u (B X ) that are uniformly weakly continuous on B X . In this paper we study a generalization of the notion of boundary for these algebras, originally introduced by Globevnik. In particular, we characterize the boundaries of A wu (B X ) when the dual of X is separable. We exhibit some natural examples of Banach spaces where this characterization provides concrete criteria for the boundary. We also show that every non-reflexive Banach space X which is an M-ideal in its bidual cannot have a minimal closed boundary for A u (B X ).