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 $\mathcal{D}$ if and only if $\mathfrak{I}$ and $\mathfrak{I}\otimes \mathcal{D}$ 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.