A Riemann manifold structure of the spectra of weighted algebras of holomorphic functions
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.
scientific article; zbMATH DE number 5706246