# Injectivity, crossed products, and amenable group actions

## TL;DR

This paper investigates when maximal and reduced crossed products of G-C*-algebras coincide, exploring their links with amenability and injectivity, and introduces new characterizations and tools for understanding these properties.

## Contribution

It provides new connections between amenability, injectivity, and crossed product properties, including complete characterizations of equivariant injectivity and weak expectation properties.

## Key findings

- Characterization of equivariant injectivity and weak expectation property
- Conditions under which maximal and reduced crossed products agree
- Connections between amenability and injectivity in G-C*-algebras

## Abstract

This paper is motivated primarily by the question of when the maximal and reduced crossed products of a $G$-$C^*$-algebra agree (particularly inspired by results of Matsumura and Suzuki), and the relationships with various notions of amenability and injectivity. We give new connections between these notions. Key tools in this include the natural equivariant analogues of injectivity, and of Lance's weak expectation property: we also give complete characterizations of these equivariant properties, and some connections with injective envelopes in the sense of Hamana.