Symmetric tensor algebras and integral decompositions
type: Item
instance of: scholarly article
title: Symmetric tensor algebras and integral decompositions
zbMATH Open document ID: 0703.46045
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: Let \(\lambda\) be a perfect sequence space closed under the Cauchy convolution product, with respect to which it is a topological algebra with jointly continuous product. Suppose, in addition, that \(\lambda\) is type (h) and \(\lambda\subset h\). Let \(E[t]\) be a nuclear Fréchet space, with the topology t defined by the seminorms \(\{p_ r^{(1)}:\) \(r\geq 0\}\). The corresponding regular tensor algebra is \[ T(E,\lambda)=\{x\in \prod_{n\geq 0}{\hat \otimes}^ n E:\;[p_ r^{(n)}(x_ n)]\in \lambda,\quad r\geq 0\}, \] equipped with the topology \(t(E,\lambda^{\times})\) determined by the seminorms \[ p_{u,r}(x)=\sum_{n\geq 0}| u_ n| p_ r^{(n)}(x_ n),\quad (u\in \lambda^{\times}). \] The algebra product is \[ (xy)_ n=\sum_{0\leq k\leq n}x_{n-k}\otimes y_ k. \] If \(S_ n\) is the symmetrizing operator on \({\hat \otimes}^ n E\) for all \(n\geq 0\), we consider the symmetric subalgebra \(S[T(E,\lambda)]\), where \(S(x_ n)=(S_ nx_ n).\) We show that functionals on \(T(E,\lambda)\) and \(S[T(e,\lambda)]\) can be represented by complete finite complex Borel measures if they are continuous with respect to a certain locally convex topology \(\hat t(E,\lambda^{\times})\) on \(T(E,\lambda)\) and its restriction \(\hat t_ S(E,\lambda^{\times})\) to \(S[T(E,\lambda^{\times})]\) respectively. The topology \(\hat t(E,\lambda^{\times})\) is coarser than \(t(E,\lambda^{\times})\) and non-Hausdorff, whereas \(\hat t_ S(E,\lambda^{\times})\) is Hausdorff. If the algebraic positive cone in \(\lambda\) is normal, then so is the positive cone on \(S[T(E,\lambda)]\). Using the isomorphism \(\lambda\cong T({\mathbb{C}},\lambda)\), we show that the cone in \(s[T(E,\lambda)]\) is normal for all nuclear Fréchet spaces \(E[t]\) if and only if \(t({\mathbb{C}},\lambda^{\times})= \hat t({\mathbb{C}},\lambda^{\times})\).
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 4153238
zbMATH Keywords: perfect sequence space closed under the Cauchy convolution product
zbMATH Keywords: topological algebra with jointly continuous product
zbMATH Keywords: nuclear Fréchet space
zbMATH Keywords: regular tensor algebra
zbMATH Keywords: symmetrizing operator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195172514
OpenAlex ID: W1966770050
cites work: I*-algebras and their applications
cites work: On structure of the algebra of field operators
cites work: Necessary and sufficient conditions for integral representations of Wightman functional at Schwinger points
cites work: Regular tensor algebras
cites work: Q5569793
cites work: Q5731609
DOI: 10.2977/PRIMS/1195172514
description: scientific article; zbMATH DE number 4153238