Symmetric continuous Reinhardt domains
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 \(\omega \in \Omega _{j(\omega )}\).
scientific article; zbMATH DE number 2051576