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)⊗Kj(B)→Ki+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 Kasparovʹs 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 separableC*-algebraA, in such a way thatKi(A) is canonically identified with the ordinaryK-theory of the dual algebra,K1−i(D(A)). Higsonʹs 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].