Regular tensor algebras
Summary: A convolution algebra is a normal sequence space \(\lambda\) that is closed under the convolution product. Let E be a nuclear Fréchet locally convex space, and let \({\mathcal P}=\{p_ r:\) \(r\in {\mathbb{N}}\cup \{0\}\}\) be a family of Hilbertian seminorms defining its topology. We define the vector space \[ T(E,\lambda,{\mathcal P})=\{(x_ n)\in \prod_{n\geq 0}{\hat \otimes}^ n E:\;(\otimes^ n_{\sigma} p_ r(x_ n))\in \lambda,\quad r\geq 0\}, \] equipped with the topology obtained from the seminorms \[ \sum_{n\geq 0}| u_ n| \otimes^ n_{\sigma} p_ r, \] where \(r\geq 0\) and \(u\in \lambda^{\times}\), the Köthe dual of \(\lambda\). For certain sequence spaces, the resulting space does not depend on \({\mathcal P}\) and we write T(E,\(\lambda\)). Such is the case for sequence spaces of type h. Certain properties of \(\lambda\) transfer directly to T(E,\(\lambda\),\({\mathcal P})\). In particular, if \(\lambda\) is complete (respectively Fréchet, a topological algebra), then so is T(E,\(\lambda\),\({\mathcal P})\). A regular tensor algebra is a space T(E,\(\lambda\)) for \(\lambda\) a perfect topological convolution algebra with jointly continuous product which is of type h. T(E,\(\lambda\)) is then nuclear, and reflexive if Fréchet. We examine the topological properties of the spaces T(E,\(\lambda\),\({\mathcal P})\). Other than invertibility, these are the same as for \(E_{\otimes}={\bar \oplus}_{n\geq 0}\otimes^ n E.\) We then consider the order properties arising from a natural involution. The positive wedge K(\(\lambda\)) is always a proper strict-b cone, and if \(\lambda\) is of type h and \(\lambda\subseteq h\), its closure is proper. Here h is the sequence space isomorphic to the space H(\({\mathbb{C}})\) of entire functions. In particular, T(E,h) is a regular tensor algebra which is locally multiplicatively convex and whose closed cone is proper. Finally, we present three conditions which are sufficient for K(\(\lambda\)) to be normal.
scientific article; zbMATH DE number 4153237
scientific article