# Lifting theorems for completely positive maps

### James Gabe

University of Southern Denmark, Odense, Denmark

## 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 $E(X;A,B)≅KK(X;A,B)$ 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 $A⊗O_{2}≅A$ if and only if every two-sided, closed ideal in $A$ is $KK$-equivalent to zero.

## Cite this article

James Gabe, Lifting theorems for completely positive maps. J. Noncommut. Geom. 16 (2022), no. 2, pp. 391–421

DOI 10.4171/JNCG/479