Equivariant injectivity of crossed products

TL;DR

This paper develops a framework for studying $G$-operator spaces and their crossed products, establishing conditions for $G$-injectivity related to amenability and generalizing recent results in quantum group theory.

Contribution

It introduces $G$-operator spaces and analyzes their crossed products, providing new characterizations of $G$-injectivity and unifying previous research in quantum group operator space theory.

Findings

Crossed product $X times_G G$ is $G$-injective iff it is injective and $G$ is amenable.

$X times_G G$ is $reve{G}$-injective iff $X$ is $G$-injective.

Results generalize and unify recent literature in quantum operator spaces.

Abstract

We introduce the notion of a $\mathbb{G}$-operator space $(X, \alpha)$, which consists of an action $\alpha: X \curvearrowleft \mathbb{G}$ of a locally compact quantum group $\mathbb{G}$ on an operator space $X$, and we make a study of the notion of $\mathbb{G}$-equivariant injectivity for such an operator space. Given a $\mathbb{G}$-operator space $(X, \alpha)$, we define a natural associated crossed product operator space $X\rtimes_\alpha \mathbb{G}$, which has canonical actions $X\rtimes_\alpha \mathbb{G} \curvearrowleft \mathbb{G}$ (the adjoint action) and $X\rtimes_\alpha \mathbb{G}\curvearrowleft \check{\mathbb{G}}$ (the dual action) where $\check{\mathbb{G}}$ is the dual quantum group. We then show that if $X$ is a $\mathbb{G}$-operator system, then $X\rtimes_\alpha \mathbb{G}$ is $\mathbb{G}$-injective if and only if $X\rtimes_\alpha \mathbb{G}$ is injective and $\mathbb{G}$ is amenable, and that (under a mild assumption) $X\rtimes_\alpha \mathbb{G}$ is $\check{\mathbb{G}}$-injective if and only if $X$ is $\mathbb{G}$-injective. We discuss how these results generalise and unify several recent results from the literature, and give new applications of these results.