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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.

46L85 Noncommutative topology
47C15 Linear operators in \(C^*\)- or von Neumann algebras
46L05 General theory of \(C^*\)-algebras