# zbMATH — the first resource for mathematics

The corona factorization property and approximate unitary equivalence. (English) Zbl 1111.46050
Let $$A$$ and $$B$$ be the stable and separable $$C^*$$-algebras, $$B$$ being an absorbing algebra, i.e., it has the corona factorization property (every norm-full extension of $$B$$ by a separable $$C^*$$-algebra is nuclearly absorbing), and $$KL(A,B)$$ the associated Rørdam’s group. The concept of approximately unitarily equivalence used in this paper is the following: Two extensions $$\phi,\psi:A\to{\mathcal M}(B)/B$$ are said to be approximately unitarily equivalent if there exists a sequence $$\{u_n\}^\infty_{n=1}$$ of unitaries in the corona algebra of $$B$$ such that $$u_n\phi(\cdot)u^*_n$$ converges to $$\psi(\cdot)$$ pointwise. The key technical result is a lemma showing that approximate unitary equivalence preserves the purely large property of G. Elliott and D. Kucerovsky [Pac. J. Math. 198, No. 2, 385–409 (2001; Zbl 1058.46041)]. Using this, Rørdam’s group is characterized as a group of purely large extensions under approximate unitary equivalence. In this way, a generalization of a Kasparov’s theorem is obtained.

##### MSC:
 46L85 Noncommutative topology 47C15 Linear operators in $$C^*$$- or von Neumann algebras 46L05 General theory of $$C^*$$-algebras