A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

Abstract

$\DeclareMathOperator{\G}{\mathbb{G}}\DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Corr}{Corr}$Let $\G$ be a locally compact quantum group and $(M,\alpha )$ a $\G$-$W^{\ast }$-algebra. The object of study of this paper is the $W^{\ast }$-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 $W^{\ast }$-category. Given another $\G$-$W^{\ast }$-algebra $(N,\beta )$, we denote the category of normal $\ast$-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 $Q\circ P\cong \mathrm{id}.$ We use these functors to show that the $\G$-dynamical von Neumann algebras $(M,\alpha )$ and $(N,\beta )$ are equivariantly Morita equivalent if and only if $\Rep^{\G}(N)$ and $\Rep^{\G}(M)$ are equivalent as $\Rep(\G)$-module-$W^{\ast }$-categories. Specializing to the case where $\G$ is a compact quantum group, we prove that moreover $P\circ Q\cong \mathrm{id}$, 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.

20 pages. Author accepted version, for publication in the Journal of Algebra

Cited by in corpus (1)

• Morita theory for dynamical von Neumann algebras