On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for ∗-homomorphisms and the topology of the KK-groups which avoids entirely C∗-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner’s topology and the Brown–Salinas topology. We study the generalized RZrdam group KL(A,B) = KK(A,B)/¯0, and prove that if a separable exact residually finite dimensional C∗-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type 2∞. In particular such an embedding exists for the C∗-algebra of a second countable amenable locally compact maximally almost periodic group. © 2005 Elsevier Inc. All rights reserved.