Differentiation in modules over topological ∗-algebras

Abstract

In recent years, there has been a systematic development of the theory of A-bundles, that is, vector bundles where the field of coefficients is replaced by a topological algebra A, usually an involutive one. The above extension is justified by its applications in pure mathematics (differential topology [15]) as well as in relativistic quantum mechanics, via, e.g., the Serre-Swan theorem [22]. Bundles with coefficients in a commutative unital Banach algebra, or yet a C*-algebra, were considered by T. K. Kandelaki [6, 71, K. Fujii [S], and A. S. MiSEenko and Yu. P. Solov’ev [ 15, 161. Furthermore, A. Mallios studied A-bundles, with A being a locally m-convex (*-)algebra with unit [l&12]. Note also that certain results are still valid for more general topological algebras [ 13, Sect. 53, or even topological rings [21]. The results already obtained are mainly within the framework of topological A-bundles. The basic difficulty in the development of a theory of differential A-bundles is the lack of a suitable differential calculus in the libres, the latter being topological modules over A. One might, of course, use the existing methods of differentiation in the underlying topological vector spaces (cf. for instance [18]). However, besides the intrinsic weaknesses of these methods (regarding the higher order chain rule, continuity of differentiable maps, differentiability of the composition map and evaluation map, which are necessary for differential geometric considerations on bundles), the additional algebraic operations (scalar multiplication by elements of A, involution) should be taken into account, because they are essentially involved in the structure of A-bundles (cf. [ 11, 183). For the same reason (incompatibility with involution), the 255 0022-247X/92 55.00