A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

TL;DR

This paper establishes a categorical framework linking Morita equivalence of dynamical von Neumann algebras under quantum group actions to equivalences of their representation categories, extending classical theorems to the quantum setting.

Contribution

It introduces a categorical interpretation of Morita equivalence for dynamical von Neumann algebras with quantum group actions, generalizing the Eilenberg-Watts theorem to the quantum context.

Findings

Categories of G-equivariant correspondences and functors are canonically related.

Morita equivalence corresponds to equivalence of representation categories as module categories.

For compact quantum groups, the correspondence categories and functor categories are equivalent.

Abstract

$\DeclareMathOperator{\G}{\mathbb{G}}\DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Corr}{Corr}$Let $\G$ be a locally compact quantum group and $(M, \alpha)$ a $\G$-$W^*$-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 $(N, \beta)$, 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 $(M, \alpha)$ and $(N, \beta)$ are equivariantly Morita equivalent if and only if $\Rep^{\G}(N)$ and $\Rep^{\G}(M)$ are equivalent as $\Rep(\G)$-module-$W^*$-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.