\(\alpha\)-derivations and their norm in projective tensor products of \(\Gamma\)-Banach algebras
type: Item
instance of: scholarly article
title: \(\alpha\)-derivations and their norm in projective tensor products of \(\Gamma\)-Banach algebras
zbMATH Open document ID: 0917.46042
DOI: 10.1155/S0161171298000490
author: H. K. Nath
author: T. K. Dutta
author: Ramesh Chandra Kalita
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1999-08-02T00:00:00Z
full work available at URL: https://eudml.org/doc/48180
review text: Summary: Let \((V,\Gamma)\) and \((V', \Gamma')\) be Gamma-Banach algebras over the fields \(F_1\) and \(F_2\) isomorphic to a field \(F\) which possesses a real-valued valuation, and \((V,\Gamma)\otimes_p(V', \Gamma')\), their projective tensor product. It is shown that if \(D_1\) and \(D_2\) are \(\alpha\)-derivation and \(\alpha'\)-derivation on \((V, \Gamma)\) and \((V', \Gamma')\) respectively and \(u= \sum_i x_i\otimes y_i\) is an arbitrary element of \((V,\Gamma) \otimes_p(V', \Gamma')\), then there exists an \(\alpha\otimes \alpha'\)-derivation \(D\) on \((V,\Gamma)\otimes_p (V',\Gamma')\) satisfying the relation \[ D(u)= \sum_i[(D_1 x_i)\otimes y_i+ x_i\otimes (D_2 y_i)] \] and possessing many enlightening properties. The converse is also true under a certain restriction. Furthermore, the validity of the results \(\| D\|=\| D_1\|+\| D_2\|\) and \(\text{sp}(D)= \text{sp}(D_1)+ \text{sp}(D_2)\) are fruitfully investigated.
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 1140280
zbMATH Keywords: Gamma-Banach algebras
zbMATH Keywords: projective tensor product
MaRDI profile type: Publication
OpenAlex ID: W2043577198
description: scientific article; zbMATH DE number 1140280