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].