On a category of module bundles
type: Item
instance of: scholarly article
title: On a category of module bundles
zbMATH Open document ID: 1113.46041
DOI: 10.1007/BF02874948
publication date: 2007-02-02T00:00:00Z
review text: A (real or complex) unital topological algebra is called a Waelbroeck algebra if the set of all its invertible elements is open, and furthermore, inversion is a continuous operation on this set. In the locally convex context, these algebras are also called continuous inversion algebras (CIAs). In case the unit is missing, one can adjoin formally a unit to the algebra, and the condition of invertibility of an element \(x\) becomes translated into the existence of \(y\) such that \(x+y+xy=0\) and \(x+y+yx=0\); an element \(x\) in a non-unital topological algebra \(A\) satisfying this condition is then called quasi-invertible, \(y\) its quasi-inverse, and \(A\) is called Waelbroeck if the set of all quasi-invertible elements is open and quasi-inversion in a continuous operation on this set. The Serre-Swan theorem states that the functor of sections provides an equivalence between the category of vector bundles on a compact Hausdorff space \(X\) and the category of finitely generated projective modules over the algebra \(A\) of continuous functions on \(X\). One deduces an equality of topological and algebraic \(0\)th \(K\)-groups of \(A\). For general locally compact Hausdorff spaces, one may prefer to consider functions with compact support. But then the unit is missing, and we are in the context of non-unital Waelbroeck algebras. An additive category is called pseudo-Abelian if for any object \(E\) and any idempotent endomorphism \(\alpha:E\to E\) (``projector''), the kernel of \(\alpha\) exists. In this context, the paper under review provides a study of a category \({\mathcal E}_A(X)\) of bundles of \(A\)-modules over a non-unital Waelbroeck algebra \(A\). The aim is to generalize results from the unital or locally convex setting. The main theorem states that for a topological algebra \(A\), an \(A\)-bundle \((E,\pi,X)\) on a topological space \(X\) and a projector \(\alpha\) on \(E\), the union of kernels of \(\alpha\) is an object of \({\mathcal E}_A(X)\), provid…
reviewed by: Friedrich Wagemann
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46M15
Mathematics Subject Classification ID: 46H25
zbMATH DE Number: 5121530
zbMATH Keywords: Waelbroeck algebra
zbMATH Keywords: quasi-invertible elements
zbMATH Keywords: pseudo-Abelian additive category
zbMATH Keywords: topological algebra
author: Mart Abel
MaRDI profile type: Publication
cites work: On finitely generated projective modules
cites work: Q4104421
cites work: Q4160083
cites work: Q4064160
cites work: Vector bundles and K-theory over topological algebras
cites work: Continuous vector bundles over topological algebras
full work available at URL: https://doi.org/10.1007/bf02874948
OpenAlex ID: W1999129489
published in: Rendiconti del Circolo Matematico di Palermo
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description: scientific article; zbMATH DE number 5121530