Equivariant injectivity of crossed products

Abstract

We introduce the notion of a \(\mathbb{G}\)-operator space, 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 study the notion of \(\mathbb{G}\)-equivariant injectivity for such an operator space. 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 some new applications.