Lifting theorems for completely positive maps

James Gabe*

*Corresponding author for this work

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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 ⊗ O2 Š A if and only if every two-sided, closed ideal in A is KK-equivalent to zero.

Original languageEnglish
JournalJournal of Noncommutative Geometry
Issue number2
Pages (from-to)391-421
Publication statusPublished - 11. Sept 2022


  • ideal related KK-theory
  • Lifting completely positive maps
  • strongly self-absorbing C -algebras


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