The maximal injective crossed product

A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes_{\inj}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^*$-algebras; this is a sort of a `dual' result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum-Connes conjecture. It turns out that $\rtimes_\inj$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.