Boundaries for algebras of holomorphic functions on Marcinkiewicz sequence spaces

Abstract

Let A(b) (E) be the Banach algebra of all complex-valued bounded continuous functions on the closed unit ball B-E of a complex Banach space E and holomorphic in the interior of B-E and let A(u) (E) be the closed subalgebra of those functions which are uniformly continuous on B-E. For the case E = M-w(0) whose bidual is a Marcinkiewicz sequence space M-w, we describe some sufficient conditions for a set to be a boundary of either A(b) (E) or A(u) (E). Moreover, we consider some analogous problems on M-w(0) to those which were studied on the Gowers space G(p) of characteristic p by Grados and Moraes [L.R. Grados, L.A. Moraes, Boundaries for algebras of holomorphic functions, J. Math. Anal. Appl. 281 (2003) 575-586; L.R. Grados, L.A. Moraes, Boundaries for an algebra of bounded holomorphic functions, J. Korean Math. Soc. 41 (1) (2004) 231-242]. (c) 2005 Elsevier Inc. All rights reserved.