On topologizing invertibility
type: Item
instance of: scholarly article
title: On topologizing invertibility
zbMATH Open document ID: 0582.46052
author: A. D. Thatte
author: S. J. Bhatt
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Motivated by topological zero devisors, we say that an element x in a metrizable topological algebra (A,T) is (boundedly) topologically invertible if for some (bounded) sequence \((x_ n)\) in A, \(xx_ n\to 1\) and \(x_ nx\to 1\). If A is a complete locally m-convex algebra, or a complete locally bounded algebra or is a locally convex algebra with Montel property, then such an x in A is invertible in A. The Arens' algebra \(L^ w[0,1]=\cap_{1\leq p<\infty}L^ p[0,1]\) [\textit{R. Arens}, Bull. Am. Math. Soc. 52, 931-935 (1946; Zbl 0060.269)] provides an example of a locally convex algebra admitting a topologically invertible element which is not invertible. However, a metrizable locally convex topological algebra A can be embedded in a larger metrizable locally convex algebra \(\ell^{\infty}(A)\) (of all bounded sequences in A) such that all boundedly topologically invertible elements in A are invertible in \(\ell^{\infty}(A)\).
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 3931784
zbMATH Keywords: topological zero devisors
zbMATH Keywords: metrizable topological algebra
zbMATH Keywords: complete locally m-convex algebra
zbMATH Keywords: complete locally bounded algebra
zbMATH Keywords: locally convex algebra with Montel property
zbMATH Keywords: Arens' algebra
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3931784