Lifting theorems for completely positive maps

Abstract

We prove lifting theorems for completely positive maps going out of exact $C^\ast$-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if $\mathsf X$ is a second countable topological space, $\mathfrak A$ and $\mathfrak B$ are separable, nuclear $C^\ast$-algebras over $\mathsf X$, and the action of $\mathsf X$ on $\mathfrak A$ is continuous, then $E(\mathsf X; \mathfrak A, \mathfrak B) \cong KK(\mathsf X; \mathfrak A, \mathfrak B)$ naturally. As an application, we show that a separable, nuclear, strongly purely infinite $C^\ast$-algebra $\mathfrak A$ absorbs a strongly self-absorbing $C^\ast$-algebra $\mathscr D$ if and only if $\mathfrak I$ and $\mathfrak I\otimes \mathscr D$ are $KK$-equivalent for every two-sided, closed ideal $\mathfrak I$ in $\mathfrak A$. In particular, if $\mathfrak A$ is separable, nuclear, and strongly purely infinite, then $\mathfrak A \otimes \mathcal O_2 \cong \mathfrak A$ if and only if every two-sided, closed ideal in $\mathfrak A$ is $KK$-equivalent to zero.