Journal of Operator Theory

Volume 95, Issue 2, Spring 2026 pp. 437-480.

Equivariant injectivity of crossed products

Authors: Joeri De Ro
Author institution: Department of Mathematics and Data Science, Vrije Universiteit Brussel, Brussels, 1050, Belgium

Summary: 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} $ and $X\rtimes_\alpha \mathbb{G} \curvearrowleft \check{\mathbb{G} }$ where $\check{\mathbb{G} }$ is the dual quantum group. We 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 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.

DOI: http://dx.doi.org/10.7900/jot.2024jun10.2478
Keywords: equivariant injectivity, crossed product, operator space, operator system, locally compact quantum group, Fubini tensor product, injective envelope, quantum group, von Neumann algebra

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