Vol. 55, No. 2 DUKE MATHEMATICAL JOURNAL (C) June 1987 THE KUNNETH THEOREM AND THE UNIVERSAL COEFFICIENT THEOREM FOR KASPAROV’S GENERALIZED K-FUNCTOR JONATHAN ROSENBERG AND CLAUDE SCHOCHET 1. Introduction. Given C*-algebras A and B with modest hypotheses, G. G. Kasparov [21] has defined groups KKi(A, B) (j 0, 1) which play a fundamen- tal role in the modem theory of C*-algebras and, particularly, its application in areas related to global analysis and algebraic topology. In this paper we prove a Kiinneth Theorem which determines the Kasparov groups in terms of the periodic K-theory groups K,(B) of Karoubi and the Brown-Douglas-Fillmore groups K*(A), and we establish a Universal Coefficient Theorem (UCT) of the form 0 Ext(K,(A), K,(B)) KK,(A, B) Hom(K,(A), K,(B)) - 0 which determines the Kasparov groups in terms of K-theory. These short exact sequences are split, unnaturally. When B C (the complex numbers) we obtain [with a new, perhaps simpler, proof] the UCT of L. G. Brown [5] for the Brown-Douglas-Fillmore groups K*(A)-- KK,(A, (2). A great deal of the power of Kasparov’s theory comes from the existence of a "Kasparov intersection product" with good functorial properties, generalizing all of the usual products (cup, cap, slant, etc.) in topological K-theory. For reasons to be explained later, our UCT also determines this product structure. In particular, we determine the structure of the graded ring KK,(A, A). Our results should be useful in several situations where the KK-groups are encountered. These include the classification of extensions of C*-algebras, index theory on foliated manifolds (as in [10]), and index theory for elliptic operators with "coefficients" in a C*-algebra (as in [25] and [30], 3B). For instance, a family of elliptic pseudodifferential operators over a compact manifold M with parameter space Y defines an element of KK,(C(M), C0(Y)); hence the compu- tation of this group is of some interest. In Section 8 we discuss some applications to the "algebraic topology" of C*-algebras. The computation of the graded ring KK,(A, A) and its graded module KK,(A (R) A, A) in a few basic cases makes it possible for us to determine all of the "homology operations" and "admissible multiplications" for mod p K-theory of C*-algebras. Received March 27, 1985. Revision received June 5, 1986. Research of first author partially supported by the National Science Foundation Grant MCS-82-00706 and by the Alfred P. Sloan Foundation. Research of second author partially supported by the National Science Foundation Grant MCS-80-01857. 431