Equivariant injectivity of crossed products
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909574354698240 |
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| author | De Ro, Joeri |
| author_facet | De Ro, Joeri |
| contents | We introduce the notion of a $\mathbb{G}$-operator space $(X, α)$, which consists of an action $α: X \curvearrowleft \mathbb{G}$ 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 $(X, α)$, we define a natural associated crossed product operator space $X\rtimes_α\mathbb{G}$, which has canonical actions $X\rtimes_α\mathbb{G} \curvearrowleft \mathbb{G}$ (the adjoint action) and $X\rtimes_α\mathbb{G}\curvearrowleft \check{\mathbb{G}}$ (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 $X\rtimes_α\mathbb{G}$ is $\mathbb{G}$-injective if and only if $X\rtimes_α\mathbb{G}$ is injective and $\mathbb{G}$ is amenable, and that (under a mild assumption) $X\rtimes_α\mathbb{G}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10738 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equivariant injectivity of crossed products De Ro, Joeri Operator Algebras Group Theory Quantum Algebra We introduce the notion of a $\mathbb{G}$-operator space $(X, α)$, which consists of an action $α: X \curvearrowleft \mathbb{G}$ 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 $(X, α)$, we define a natural associated crossed product operator space $X\rtimes_α\mathbb{G}$, which has canonical actions $X\rtimes_α\mathbb{G} \curvearrowleft \mathbb{G}$ (the adjoint action) and $X\rtimes_α\mathbb{G}\curvearrowleft \check{\mathbb{G}}$ (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 $X\rtimes_α\mathbb{G}$ is $\mathbb{G}$-injective if and only if $X\rtimes_α\mathbb{G}$ is injective and $\mathbb{G}$ is amenable, and that (under a mild assumption) $X\rtimes_α\mathbb{G}$ 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. |
| title | Equivariant injectivity of crossed products |
| topic | Operator Algebras Group Theory Quantum Algebra |
| url | https://arxiv.org/abs/2312.10738 |