A Riemann manifold structure of the spectra of weighted algebras of holomorphic functions
type: Item
instance of: scholarly article
title: A Riemann manifold structure of the spectra of weighted algebras of holomorphic functions
zbMATH Open document ID: 1198.46026
author: Daniel Carando
author: Domingo García
author: Manuel Maestre
author: Pablo Sevilla-Peris
published in: Topology
publication date: 2010-05-10T00:00:00Z
review text: Given an open subset \(U\) of a complex Banach space \(X\) and a countable family \(V\) of weights, \(HV(U)\) denotes the space of all analytic functions \(f:U\to \mathbb{C}\) such that \(|f|\upsilon\) is bounded for all \(\upsilon\in V.\) It is endowed with the topology given by the family of seminorms \(\{\sup_{x\in U}|f(x)|\upsilon(x):\upsilon\in V\}\). In the case when the space is also a Fréchet algebra, \(\mathfrak{M}V(U)\) denotes the set of all continuous scalar homomorphisms on \(HV(U)\), that is, the spectrum. Continuing earlier work by \textit{R.\,M.\thinspace Aron, P.\,Galindo, D.\,García} and \textit{M.\,Maestre} [Trans.\ Am.\ Math.\ Soc.\ 348, No.\,2, 543--559 (1996; Zbl 0844.46024)], the authors provide conditions on \(V\) for \(\mathfrak{M}V(U)\) to have a Riemann manifold structure over \(X^{**}\) given by the restriction map \(\phi\in \mathfrak{M}V(U)\mapsto \phi_{|_{X^*}}.\) This is done in the setting of symmetrically regular Banach spaces \(X,\) i.e., spaces where any operator \(T:X\to X^*\) such that \(Tx(y)=Ty(x),\; x,y\in X\), is weakly compact. Those conditions are rather technical and several examples of families of weights \(V\) fulfilling them are provided in the article under review. When dealing with entire functions, that is, \(U=X,\) a slight strengthening of the given conditions allows the authors to prove that \(\mathfrak{M}V(X)\) is a disjoint union of copies of \(X^{**}\) in such a way that their embedding into the spectrum is a bianalytic (onto the range) mapping.
reviewed by: Pablo Galindo
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46J20
zbMATH DE Number: 5706246
zbMATH Keywords: weighted space of holomorphic functions
zbMATH Keywords: Fréchet algebra
zbMATH Keywords: analytic manifold structure
zbMATH Keywords: symmetrically regular Banach space
MaRDI profile type: Publication
full work available at URL: j.top.2009.11.003
OpenAlex ID: W2086403751
DOI: 10.1016/J.TOP.2009.11.003
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description: scientific article; zbMATH DE number 5706246