Regular tensor algebras
type: Item
instance of: scholarly article
title: Regular tensor algebras
zbMATH Open document ID: 0703.46044
author: Daniel A. Dubin
author: M. A. Hennings
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 4153237
zbMATH Keywords: convolution algebra
zbMATH Keywords: nuclear Fréchet locally convex space
zbMATH Keywords: family of Hilbertian seminorms
zbMATH Keywords: Köthe dual
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195172513
OpenAlex ID: W2057130742
cites work: Order properties of a class of tensor algebras
cites work: Q3699418
cites work: I*-algebras and their applications
cites work: On structure of the algebra of field operators
cites work: On locally m-convex x-algebras
cites work: Generalized B-Algebras
cites work: Q5513660
cites work: Q5674507
cites work: Q4165775
cites work: Q4746304
cites work: Q3918541
cites work: Q5569793
cites work: Topological algebras of operators
cites work: On positive functionals on algebras of test functions for quantum fields
cites work: Q5555984
cites work: Graded and filtrated topological *-algebras. I: Graded normal topologies
cites work: Q5508878
cites work: Q5557339
cites work: Q5731609
cites work: Quantum Field Theory in Terms of Vacuum Expectation Values
cites work: The field algebra and its positive linear functionals
cites work: On the algebra of test functions for field operators. Decomposition of linear functionals into positive ones
DOI: 10.2977/PRIMS/1195172513
description: scientific article; zbMATH DE number 4153237