The maximal injective crossed product

Abstract

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$A\mapsto A\rtimes _{\text{inj}}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of$G$-injective$C^{\ast }$-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$\rtimes _{\text{inj}}$has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.