The maximal injective crossed product

A crossed product functor is said to beinjectiveif 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↦ A ᵢₙⱼG . 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 ‘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 ᵢₙⱼ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.