Fundamental to the analytic K-homology theory of G. Kasparov [7, is the construction of the external product in K-homology
This construction is modeled on the ``sharp product'' of elliptic operators over compact manifolds , and involves some deep functional-analytic considerations which at first sight may appear somewhat ad hoc.
A different approach to Kasparov's theory has recently been expounded by N. Higson [5], following the lead of W. Paschke . He constructs a dual algebra'' D(A) for any separable C\*-algebra A, in such a way that K i (A) is canonically identified with the ordinary K-theory of the dual algebra, K 1&i (D(A)). Higson's treatment covers the exactness and excision properties of K-homology, but stops short of the Kasparov product; it is natural to ask whether the product itself can be given a dual'' interpretation, in terms of the external product in ordinary K-theory. It is the purpose of this article to show that this can indeed be done.
A more leisurely exposition of K-theory and K-homology from this perspective will appear in .
1998 Academic Press
1. K-THEORY OF GRADED C*-ALGEBRAS
In this section we briefly review and reformulate a construction of K-theory for graded C*-algebras, due to van Daele . We will consider ZΓ2graded complex C*-algebras. The whole discussion also applies to the real case with only minor changes.
(1.1) Definition. A supersymmetry in a graded, unital C*-algebra is an odd, self-adjoint, unitary element. We let SS(A) denote the space of supersymmetries in A; it is equipped with the topology induced by the C*-norm.
We will confine our attention to those unital C*-algebras which contain supersymmetries. From the point of view of K-theory this is no loss, as there is a simple stabilization (see ) which converts any given C*-algebra to a K-theoretically equivalent one containing supersymmetries. In fact, this stabilization amounts to the (graded) tensor product with the Clifford article no. FU973224