The maximal injective crossed product

TL;DR

This paper introduces a maximal injective crossed product functor for all locally compact groups, providing an explicit construction and exploring its connections to key properties like exactness and amenability.

Contribution

It constructs the first maximal injective crossed product functor for any locally compact group, expanding the understanding of crossed product functors in operator algebras.

Findings

Existence of a maximal injective crossed product for all locally compact groups

Explicit construction depending on maximal crossed product and G-injective C*-algebras

Connections to exactness, local lifting property, and amenability

Abstract

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.