IUMJ

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

Authors: Maria D. Acosta, Richard M. Aron and Luiza A. Moraes

Issue: Volume 58 (2009), Issue 6, 2575-2596

Abstract:

For a complex Banach space $X$, let $\mathcal{A}_{u}(B_X)$ 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 $\mathcal{A}_{wu}(B_X)$ be the Banach subalgebra consisting of those functions in $\mathcal{A}_{u}(B_X)$ that are uniformly weakly continuous on $B_{X}$. In this paper we study a generalization of the notion of \emph{boundary} for these algebras, originally introduced by Globevnik. In particular, we characterize the boundaries of $\mathcal{A}_{wu}(B_X)$ 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 $\mathcal{A}_{u}(B_X)$.