One-parameter continuous fields of Kirchberg algebras. (English) Zbl 1146.46037
A Kirchberg algebra is a separable nuclear purely infinite simple \(C^*\)-algebra. The authors show that a separable, unital or stable, continuous field over \([0,1]\) of Kirchberg algebras satisfying the UCT and having finitely generated \(K\)-theory groups is isomorphic to a trivial field iff the associated \(K\)-theory presheaf is trivial. It is also shown that the \(K_d\)-sheaf, \(d\in\mathbb Z/2\mathbb Z\), is a complete invariant for separable stable continuous fields of Kirchberg algebras if the \(K_d\)-groups of the fibers are torsion free and the \(K_{d+1}\)-groups are trivial. As a corollary, it is shown that unital separable continuous fields of \(C^*\)-algebras over \([0,1]\) with fibers isomorphic to the Cuntz algebra \(\mathcal O_n\) (\(2\leq n\leq\infty\)) are trivial.
Reviewer: Vladimir M. Manuilov (Moskva)
MSC:
| 46L35 | Classifications of \(C^*\)-algebras |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
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