The corona factorization property and approximate unitary equivalence To appear at the Houston

Abstract. We study Rørdam’s group, KL(A,B), and a corona factoriza-tion condition. Our key technical result is a lemma showing that approxi-mate unitary equivalence preserves the purely large property of Elliott and Kucerovsky [10]. Using this, we characterize KL(A,B) as a group of purely large extensions under approximate unitary equivalence, generalizing a the-orem of Kasparov’s. Then we prove the following: Let B be a stable and separable C∗-algebra. Then the following are equivalent (for absorption of weakly nuclear extensions): i) The corona algebra of B has a certain quasi-invertibility property, which we here call the corona factorization property. ii) Rørdam’s group KL1nuc(A,B) is isomorphic to the group of full essen-tial extensions of A by B. iii) Every strongly full and positive element of the corona algebra of B is properly infinite. iv) Every norm-full extension of B is absorbing, with respect to approxi-mate unitary equivalence. v) Every norm-full extension of B is absorbing, with respect to ordinary unitary equivalence. vi) Every norm-full extension of B is absorbing, with respect to weak equivalence. vii) Every norm-full trivial extension of B is absorbing, with respect to unitary equivalence. viii) A K-theoretical uniqueness result for maps into M(B)/B. We show that if X is the infinite Cartesian product of spheres, then C(X)⊗K does not have the corona factorization property. We apply our technical lemma to study quasidiagonality and weak qua-sidiagonality of extensions