Kasparov Products and Dual Algebras

Fundamental to the analyticK-homology theory of G. Kasparov [7, 8] is the construction of the external product inK-homologyKⁱ(A)⊗K^j(B)→K ^i+j(A⊗B).This construction is modeled on the "sharp product" of elliptic operators over compact manifolds [2], and involves some deep functional-analytic considerations which at first sight may appear somewhatad hoc. A different approach to Kasparov's theory has recently been expounded by N. Higson [5], following the lead of W. Paschke [9]. He constructs a "dual algebra" D(A) for any separable C*-algebraA, in such a way thatKⁱ(A) is canonically identified with the ordinaryK-theory of the dual algebra,K_1-i(D(A)). Higson's treatment covers the exactness and excision properties ofK-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 ordinaryK-theory. It is the purpose of this article to show that this can indeed be done. A more leisurely exposition ofK-theory andK-homology from this perspective will appear in [6].