On a category of module bundles (Q862009)

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 \({\mathcal E}_A(X)\), provided that the algebra of endomorphisms of the fiber module of \(E\) with respect to a fixed atlas is Waelbroeck. The author deduces from this theorem that \({\mathcal E}_A(X)\) is pseudo-Abelian for \(A\) Waelbroeck, using that the endomorphism algebras of any topological module over a Waelbroeck algebra are Waelbroeck. The paper is sufficiently detailed and written in a clear style.