Vector bundles and K-theory over topological algebras
Let \({\mathbb{A}}\) be a locally given m-convex algebra with an identity element. In part 1 a topological \({\mathbb{A}}\)-module is defined. The category \({\mathcal P}({\mathbb{A}})\) of finitely generated projective \({\mathbb{A}}\)- modules with morphisms the \({\mathbb{A}}\)-linear maps is formed and in section 1-3 \({\mathbb{A}}\)-vector bundles are systematically examined. Let X be a topological space and \({\mathbb{A}}^ a \)topological algebra. In part 3 the category \({\mathcal E}_{{\mathbb{A}}}(X)\) is defined whose objects are the \({\mathbb{A}}\)-bundles over X. In part 4 the classical result for complex vector bundles due to J.-P. Serre and R. G. Swan [see \textit{M. Karoubi}, K-theory (1978; Zbl 0382.55002), p. 32] is summarized. Let \(K_{{\mathbb{A}}}(X)\) be the Grothendieck group of the additive category \({\mathcal E}_{{\mathbb{A}}}(X)\), i.e. \(K_{{\mathbb{A}}}(X):=K({\mathcal E}_{{\mathbb{A}}}(X))\). Among the results obtained in part 5 we shall point out the vectorization of the formula concerning ''Bott periodicity'' (Theorem 5.2). Let (X,\({\mathcal A})\) be a topological geometric space, i.e. X is any Hausdorff topological space and \({\mathcal A}^ a \)sheaf of topological algebras on X (part 6). In part 7 for each pair (X,Y) consisting of a compact space X and a closed subspace \(Y\subset X\), the author considers the category \({\mathcal E}_{{\mathbb{A}}}^{p,q}(X,Y)\) and designates by \(K_{{\mathbb{A}}}^{p,q}(X,Y)\) the relativized Grothendieck group of this category. In Definition 7.1. the Chern character for an element of \(\oplus_{p,q\geq 0}K_{{\mathbb{A}}}^{p,q}(X,Y)\) is defined. These formulae are analogous to the Chern character of complex vector bundles. The paper ends up with a large number of references (82 titles).
scientific article; zbMATH DE number 3899137
scientific article