Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2004; 41(1): 231-242

Printed January 1, 2004

Copyright © The Korean Mathematical Society.

Boundaries for an algebra of bounded holomorphic functions

L. A. Moraes and L. Romero Grados

Universidade Federal do Rio de Janeiro, Universidade Estadual de Ponta Grossa

Abstract

Let $A_b(B_E)$ 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 $B_E$, endowed with the sup norm. We present some sufficient conditions for a set to be a boundary for $A_b(B_E)\;$ in case $E$ belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in [6]. We also prove the non-existence of the Shilov boundary for $A_b(B_E)$ and give some examples of boundaries.

Keywords: Banach algebra, boundary, holomorphic functions

MSC numbers: Primary 46J15, 46B45; Secondary 46G20, 32A40