Equivariant bundles and absorption

For a locally compact group $G$ and a strongly self-absorbing $G$-algebra $(\mathcal{D},\delta)$, we obtain a new characterization of absorption of a strongly self-absorbing action using almost equivariant completely positive maps into the underlying algebra. The main technical tool to obtain this characterization is the existence of almost equivariant lifts for equivariant completely positive maps, proved in recent work of the authors. This characterization is then used to show that an equivariant $C_0(X)$-algebra with $\mathrm{dim}_{\mathrm{cov}}(X)<\infty$ is $(\mathcal{D},\delta)$-stable if and only if all of its fibers are, extending a result of Hirshberg, R{\o}rdam and Winter to the equivariant setting. The condition on the dimension of $X$ is known to be necessary, and we show that it can be removed if, for example, the bundle is locally trivial.