Equivariant injectivity of crossed products

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 $X{⋊}_{\alpha }\mathbb{G}$, which has canonical actions $X{⋊}_{\alpha }\mathbb{G}\curvearrowleft \mathbb{G}$ (the adjoint action) and $X{⋊}_{\alpha }\mathbb{G}\curvearrowleft \check{\mathbb{G}}$ (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 $X{⋊}_{\alpha }\mathbb{G}$ is $\mathbb{G}$-injective if and only if $X{⋊}_{\alpha }\mathbb{G}$ is injective and $\mathbb{G}$ is amenable, and that (under a mild assumption) $X{⋊}_{\alpha }\mathbb{G}$ 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.

33 pages. Author accepted version, for publication in Journal of Operator Theory