Vector bundles and K-theory over topological algebras

Abstract

Functional analysis techniques, mainly pertaining to the theory of intinitedimensional representations, have been recently applied, mostly by Soviet mathematicians, to compute Hermitian K-groups related to the classification of quadratic forms with coefftcients in group rings (see, for instance, the relevant survey articles by MiSEenko [59, 801. In this concern, it has been pointed out that without loss of generality, one may profitably consider instead as “coefficient ring” an (abstract) C*-algebra [59, p. 931. In a similar framework, namely, that of (vector) bundles with coefftcients in C*algebras, topological invariants of elliptic operators related to the classical Atiyah-Singer formula have also been studied by MiiEenko in [60] (see also [61 I), whereas, Kasparov has given in [42] a similar study by still applying C*-algebra techniques (cf. the comments at the end of this paper). Besides, vector bundles over Banach algebras and, in particular C*algebras, commutative or not, have already been applied by Kandelaki [37, 381, and independently by Fujii [27], one of their main results being a generalization of the Serre-Swan theorem [41] for complex vector bundles (cf. Theorem 4.2 in this paper). Furthermore, connections with the theory of “extensions of C*-algebras” ([ 12, “BDF-Theory”]) have also been considered by Kandelaki [39, 401. On the other hand, topological (in particular, locally m-convex) *-algebras (necessarily not normed ones) have been recently used in connection, for instance, with applications to relativistic quantum theory [18, 191. Thus, the extension of the preceding considerations to vector bundles with coefftcients in topological algebras, more general than normed ones and, in particular, in locally m-convex ones [55, 581 is naturally revealed. Now, this task is further feasible as a consequence of the generalization, within the topological algebras theory framework, of previous relevant work concerning applications of algebraic topology and complex analysis to Banach algebras theory [24, 53-55, 621. 452 0022-247X/83$3.00