A categorical interpretation of Morita equivalence for dynamical von Neumann algebras
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910767968681984 |
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| author | De Ro, Joeri |
| author_facet | De Ro, Joeri |
| contents | $\DeclareMathOperator{\G}{\mathbb{G}}\DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Corr}{Corr}$Let $\G$ be a locally compact quantum group and $(M, α)$ a $\G$-$W^*$-algebra. The object of study of this paper is the $W^*$-category $\Rep^{\G}(M)$ of normal, unital $\G$-representations of $M$ on Hilbert spaces endowed with a unitary $\G$-representation. This category has a right action of the category $\Rep(\G)= \Rep^{\G}(\mathbb{C})$ for which it becomes a right $\Rep(\G)$-module $W^*$-category. Given another $\G$-$W^*$-algebra $(N, β)$, we denote the category of normal $*$-functors $\Rep^{\G}(N)\to \Rep^{\G}(M)$ compatible with the $\Rep(\G)$-module structure by $\operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ and we denote the category of $\G$-$M$-$N$-correspondences by $\operatorname{Corr}^{\G}(M,N)$. We prove that there are canonical functors $P: \Corr^{\G}(M,N)\to \operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ and $Q: \operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))\to \operatorname{Corr}^{\G}(M,N)$ such that $Q \circ P\cong \operatorname{id}.$ We use these functors to show that the $\G$-dynamical von Neumann algebras $(M, α)$ and $(N, β)$ are equivariantly Morita equivalent if and only if $\Rep^{\G}(N)$ and $\Rep^{\G}(M)$ are equivalent as $\Rep(\G)$-module-$W^*$-categories. Specializing to the case where $\G$ is a compact quantum group, we prove that moreover $P\circ Q \cong \operatorname{id}$, so that the categories $\Corr^{\G}(M,N)$ and $\operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A categorical interpretation of Morita equivalence for dynamical von Neumann algebras De Ro, Joeri Operator Algebras Category Theory Quantum Algebra $\DeclareMathOperator{\G}{\mathbb{G}}\DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Corr}{Corr}$Let $\G$ be a locally compact quantum group and $(M, α)$ a $\G$-$W^*$-algebra. The object of study of this paper is the $W^*$-category $\Rep^{\G}(M)$ of normal, unital $\G$-representations of $M$ on Hilbert spaces endowed with a unitary $\G$-representation. This category has a right action of the category $\Rep(\G)= \Rep^{\G}(\mathbb{C})$ for which it becomes a right $\Rep(\G)$-module $W^*$-category. Given another $\G$-$W^*$-algebra $(N, β)$, we denote the category of normal $*$-functors $\Rep^{\G}(N)\to \Rep^{\G}(M)$ compatible with the $\Rep(\G)$-module structure by $\operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ and we denote the category of $\G$-$M$-$N$-correspondences by $\operatorname{Corr}^{\G}(M,N)$. We prove that there are canonical functors $P: \Corr^{\G}(M,N)\to \operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ and $Q: \operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))\to \operatorname{Corr}^{\G}(M,N)$ such that $Q \circ P\cong \operatorname{id}.$ We use these functors to show that the $\G$-dynamical von Neumann algebras $(M, α)$ and $(N, β)$ are equivariantly Morita equivalent if and only if $\Rep^{\G}(N)$ and $\Rep^{\G}(M)$ are equivalent as $\Rep(\G)$-module-$W^*$-categories. Specializing to the case where $\G$ is a compact quantum group, we prove that moreover $P\circ Q \cong \operatorname{id}$, so that the categories $\Corr^{\G}(M,N)$ and $\operatorname{Fun}_{\Rep(\G)}(\Rep^{\G}(N), \Rep^{\G}(M))$ are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras. |
| title | A categorical interpretation of Morita equivalence for dynamical von Neumann algebras |
| topic | Operator Algebras Category Theory Quantum Algebra |
| url | https://arxiv.org/abs/2408.07701 |