I am currently reading through Claire Anantharaman-Delaroche's paper "Amenability and exactness for dynamical systems and their $C^*$-algebras". Arxiv-link: https://arxiv.org/abs/math/0005014.
In the proof of Theorem $4.6 \ (1)\Rightarrow (2)$ it is mentioned that the image of an approximate identity is an approximate identity of multipliers. How is this realized?
Thank you
Added details: Let $X$ be a compact Hausdorff space and let $G$ denote a locally compact Hausdorff group. Given a closed two-sided ideal $I$ in a $G$-$C^*$-algebra $A$, we denote by $\phi_I$ the $*$-homomorphism induced to the reduced crossed products by the natural inclusion of $I$ into $C(X,I)$, i.e. $\phi_I: I \rtimes_r G \to C(X,I)\rtimes_r G$.
The following quote from the paper is causing me troubles: "Given an approximate unit $(u_\lambda)$ in $I$, the net $(\phi_I(u_\lambda))$ is an approximate unit of multipliers in $C(X,I)\rtimes_r G$".