Symmetric continuous Reinhardt domains
type: Item
instance of: scholarly article
title: Symmetric continuous Reinhardt domains
zbMATH Open document ID: 1045.32025
published in: Archiv der Mathematik
publication date: 2004-03-07T00:00:00Z
review text: A continuous Reinhardt domain over a locally compact Hausdorff space \(\Omega \) is a bounded domain \(D\subset {\mathcal C}_0(\Omega )\) such that \[ f\in D,\;| g| \leq | f| \Rightarrow g\in D \quad (f,g\in {\mathcal C}_0(\Omega )) \] (\({\mathcal C}_0(\Omega )\) denotes the space of continuous functions on \(D\) which vanish at infinity). The authors show that, if the continuous Reinhardt domain \(D\) is symmetric, then it can be written as \[ D=\biggl\{f\in {\mathcal C}_0(\Omega )\mid \sup _{j\in J}\sum _{\omega \in \Omega _j} m(\omega )| f(\omega )| ^2<1\biggr\}, \] where \((\Omega _j)_{j\in J}\) is a partition of \(\Omega \) with \(\Omega _j\) finite, and \(m\) is a positive function defined on \(\Omega \) with \(0<m_0\leq m(\omega )\leq m_1<\infty \). In the proof the authors use that, according to a result by Kaup, there is a \(JB^*\)-triple structure on \({\mathcal C}_0(\Omega )\) associated with the bounded symmetric domain \(D\). The associated \(JB^*\)-triple product is shown to be given by \[ \{f,g,h\}(\omega ) =\sum _{\eta \in \Omega _{j(\omega )}} {m(\eta )\over 2} \bigl(f(\eta )\overline{g(\eta )}h(\omega ) +h(\eta )\overline{g(\eta )}f(\omega )\bigr), \] for \(f,g,h\in {\mathcal C}_0(\Omega )\), where \(j(\omega )\in J\) denotes the unique index such that \(\omega \in \Omega _{j(\omega )}\).
reviewed by: Jacques Faraut
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 2051576
zbMATH Keywords: symmetric domain
zbMATH Keywords: Reinhardt domain
author: László Stachó
author: Borut Zalar
MaRDI profile type: Publication
full work available at URL: s00013-003-4659-3
OpenAlex ID: W2013962277
DOI: 10.1007/S00013-003-4659-3
description: scientific article; zbMATH DE number 2051576