Boundaries for algebras of holomorphic functions.
type: Item
instance of: scholarly article
title: Boundaries for algebras of holomorphic functions.
zbMATH Open document ID: 1037.46051
DOI: 10.1016/S0022-247X(03)00150-1
author: Luiza A. Moraes
author: Luis Romero Grados
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(A_u(B_E)\) be the Banach algebra of all complex valued functions which are uniformly continuous on the closed unit ball \(B_E\) of a Banach space \(E\) and holomorphic in the interior of \(B_E\). The authors study the boundary and the Shilov boundary of \(A_u(B_E)\) for \(E=G_p\), the Gowers sequence space with characteristic \(p\), and for its \(n\)-dimensional version \(E=G_{p,n}\) (\(p=1\) gives the classical Gowers space). It is shown that the Shilov boundary of \(A_u(B_{G_{p,n}})\) coincides with both, the set of all extreme points of \(B_{G_{p,n}}\) and the set of all of its peak points. An explicit characterization of this set is also given. For the infinite dimensional case \(G_p\), a characterization of the boundaries in terms of its finite dimensional projections to \(G_{p,n}\) is given. Using this, it is shown that \(A_u(G_p)\) does not have a Shilov boundary for all \(p \in \mathbb N\). Finally, examples of ``small'' boundaries are constructed.
reviewed by: Heinz Junek
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 1956883
zbMATH Keywords: Banach algebras
zbMATH Keywords: boundary
zbMATH Keywords: holomorphic functions
zbMATH Keywords: Gowers space
zbMATH Keywords: peak points
zbMATH Keywords: extreme points
MaRDI profile type: Publication
cites work: Q4283362
cites work: A minimal boundary for function algebras
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cites work: Boundaries for polydisc algebras in infinite dimensions
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cites work: Q4774586
full work available at URL: s0022-247x(03)00150-1
OpenAlex ID: W2059683071
description: scientific article; zbMATH DE number 1956883