Amenability and covariant injectivity of locally compact quantum groups

TL;DR

This paper explores the relationship between amenability and covariant injectivity in locally compact quantum groups, extending known results for groups to the quantum setting through module actions.

Contribution

It establishes that the equivalence between amenability and injectivity persists for all locally compact quantum groups when considering a natural module action.

Findings

Equivalence holds for all locally compact groups with a specific module perspective.

Extension of the equivalence to the quantum group setting.

Provides a generalized framework for amenability and injectivity.

Abstract

As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if $\mathcal{L}(G)$ is considered as a $\mathcal{T}(L_2(G))$-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.