On the bamboo-shoot topology of certain inductive limits of topological groups
type: Item
instance of: scholarly article
title: On the bamboo-shoot topology of certain inductive limits of topological groups
zbMATH Open document ID: 0948.22002
author: Takashi Edamatsu
published in: Journal of Mathematics of Kyoto University
publication date: 2000-11-05T00:00:00Z
review text: Let \(\{(G_n,\tau_n),\phi_{n+1\: n}\}_{n\in \mathbb{N}}\) be an inductive system of topological groups, where \(\phi_{n+1\: n}:(G_n,\tau_n)\to(G_{n+1},\tau_{n+1})\) is a continuous homomorphism. It was shown by \textit{N. Tatsuuma, H. Shimomura} and \textit{T. Hirai} [J. Math. Kyoto Univ. 38, No. 3, 551-578 (1998; Zbl 0930.22002)] that the inductive limit topology \(\tau_{\text{ind}}\) on \(G =\varinjlim G_{n}\) is not necessarily a group topology but that under some mild conditions (the ``PTA condition'', in their terminology) a group topology can be defined on \(G\) so that every canonical map \(\phi_n :G_n \rightarrow G\) is continuous. They called the finest among such topologies the bamboo-shoot topology, \(\tau_{X_{\text{OPEN}}}\) for short. The present paper studies the topology \(\tau_{X_{\text{OPEN}}}\) for inductive systems of groups obtained from inductive limits of Banach algebras. These groups can be described in a concrete fashion as follows. Let \(\{A_ n, \psi_{n+1: n}\}_{n\in \mathbb{N}}\) be a strict inductive limit of Banach algebras with identity and assume further that its inductive limit \(A\) admits an identity. Next, let \(G(A_n)\) denote the group of invertible elements of \(A_n\) equipped with the norm topology of \(A_n\). Then \(\{G(A_n),\psi_{n+1 : n}\}_{n \in \mathbb{N}}\) is an inductive system of topological groups and its direct limit coincides, as a set, with the group of invertible elements of \(A\), \(G(A)\). On this set \(G(A)\) three topologies can be naturally brought into consideration: \(\tau_{X_{\text{OPEN}}}\) and \(\tau_{\text{ind}}\) arise when \(G(A)\) is regarded as an inductive limit of topological groups and \(\tau_{\text{lct}}\) arises from regarding \(G(A)\) as a subset of the locally convex algebra \(A\) (recall that \(A=\varinjlim A_{n}\)). The main results of this paper prove that \(\tau_{X_{\text{OPEN}}}\) coincides with \(\tau_{\text{lct}}\) on \(G(A)\) and that \(\tau_{\text{ind}}\) is a group topology if and on…
reviewed by: Jorge Galindo
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 46H99
zbMATH DE Number: 1471429
zbMATH Keywords: inductive limit
zbMATH Keywords: topological group
zbMATH Keywords: bamboo-shoot topology
zbMATH Keywords: invertible elements
zbMATH Keywords: Banach algebra
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250517824
OpenAlex ID: W1491948923
DOI: 10.1215/KJM/1250517824
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description: scientific article; zbMATH DE number 1471429