## Abstract

We prove lifting theorems for completely positive maps going out of exact C*-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if X is a second countable topological space, A and B are separable, nuclear C -algebras over X, and the action of X on A is continuous, then E.XI A; B/ Š KK.XI A; B/ naturally. As an application, we show that a separable, nuclear, strongly purely infinite C* -algebra A absorbs a strongly self-absorbing C* -algebra D if and only if I and I ⊗ D are KK-equivalent for every two-sided, closed ideal I in A. In particular, if A is separable, nuclear, and strongly purely infinite, then A ⊗ O_{2} Š A if and only if every two-sided, closed ideal in A is KK-equivalent to zero.

Originalsprog | Engelsk |
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Tidsskrift | Journal of Noncommutative Geometry |

Vol/bind | 16 |

Udgave nummer | 2 |

Sider (fra-til) | 391-421 |

ISSN | 1661-6952 |

DOI | |

Status | Udgivet - 11. sep. 2022 |

### Bibliografisk note

Funding Information:Funding. This work was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

Publisher Copyright:

© 2022 European Mathematical Society Published by EMS Press.