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 Rørdam group KL(A, B) = KK(A, B)/0̄, 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.
Dadarlat, M. (2005). On the topology of the Kasparov groups and its applications. Journal of Functional Analysis, 228(2), 394–418. https://doi.org/10.1016/j.jfa.2005.02.015