J. Korean Math. Soc. 2004; 41(1): 231-242
Printed January 1, 2004
Copyright © The Korean Mathematical Society.
L. A. Moraes and L. Romero Grados
Universidade Federal do Rio de Janeiro, Universidade Estadual de Ponta Grossa
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
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