An Algebraic Description of Boundary Maps Used in Index Theory

Abstract

In index theory and in noncommutative geometry one often associates C∗algebras with geometric objects. These algebras can for instance arise from pseudodifferential operators, differential forms, convolution algebras etc.. However they are often given a priori as locally convex algebras and one looses a certain amount of information by passing to the C∗-algebra completions. In some cases, for instance for algebras containing unbounded differential operators, there is in fact no C∗-algebra that accommodates them. On the other hand, it seems that nearly all algebraic structures arising from differential geometry can be described very naturally by locally convex algebras (or by the slightly more general concept of bornological algebras). The present note can be seen as part of a program in which we analyze constructions, that are classical in K-theory for C∗-algebras and in index theory, in the framework of locally convex algebras. Since locally convex algebras have, besides their algebraic structure, only very little structure, all arguments in the study of their K-theory or their cyclic homology have to be essentially algebraic (thus in particular they also apply to bornological algebras). This paper is triggered by an analysis of the proof of the Baum-DouglasTaylor index theorem, [2], in the locally convex setting. Consider the extension