Continuous vector bundles over topological algebras (Q1112301)

The author extends in [J. Math. Anal. Appl. 92, 452-506 (1983; Zbl 0564.18011)] classical results for complex vector bundles and K-theory to the framework of locally m-convex topological *-algebras with an identity element. The present paper improves the situation there, as for the quoted algebras the respective Q-algebras are precisely the Waelbroeck ones, given that the inversion is always a continuous map in this setting [see also the author's monograph on Topological algebras-selected topics (1986; Zbl 0597.46046)]. It becomes then clear that for a given compact Hausdorff X and a Waelbroeck algebra A, the function algebra \({\mathcal C}_ u(X,A)\) endowed with the topology of uniform convergence in X is still of the same type, a fact which is also deduced from an early result of [\textit{I. Kaplansky}, Am. J. Math. 69, 153-183 (1947; Zbl 0034.166)]. This leads again to a substantial result concerning the category \({\mathcal E}_ A(X)\) of (continuous, viz. topological) vector bundles over X with a finitely generated projective A-module as fiber i.e., it is an additive Waelbroeck category. Among other mentioned applications of it, a variant of Chern character is defined for the objects of \({\mathcal E}_ A(X)\), or more generaly, of the group \(K_ A^{*,*}(X,Y)\) for a suitable pair (X,Y) of finite CW-complexes.