A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

TL;DR

The paper develops a categorical framework for equivariant Morita theory of dynamical von Neumann algebras under a locally compact quantum group $\mathbb{G}$. It introduces equivariant correspondences $\text{Corr}^{\mathbb{G}}(M,N)$ and equivariant representation categories $\text{Rep}^{\mathbb{G}}(M)$, and constructs canonical functors $P$ and $Q$ between them, with $Q\circ P\cong \mathrm{id}$ in general and $P\circ Q\cong \mathrm{id}$ in the compact case. It proves that $\mathbb{G}$-$W^*$-Morita equivalence of $(M,\alpha)$ and $(N,\beta)$ is equivalent to $\text{Rep}^{\mathbb{G}}(M)$ and $\text{Rep}^{\mathbb{G}}(N)$ being equivalent as $\text{Rep}(\mathbb{G})$-module $W^*$-categories, and for compact $\mathbb{G}$, establishes an equivariant Eilenberg–Watts theorem: every normal $\text{Rep}(\mathbb{G})$-module functor arises from tensoring with a $\mathbb{G}$-$M$-$N$-correspondence. The results connect equivariant and non-equivariant perspectives via crossed products, yielding a robust categorical classification of equivariant Morita theory with practical implications for representations of quantum group actions on von Neumann algebras.

Abstract

$\DeclareMathOperator{\G}{\mathbb{G}}\DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Corr}{Corr}$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.

Theorems & Definitions (29)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Example 3.4
• Theorem 3.5