Equivariant Eilenberg-Watts theorems for locally compact quantum groups

Abstract

Given two von Neumann algebras $A$ and $B$, the $W^*$-algebraic Eilenberg-Watts theorem, due to M. Rieffel, asserts that there is a canonical equivalence $\operatorname{Corr}(A,B)\simeq \operatorname{Fun}(\operatorname{Rep}(B), \operatorname{Rep}(A))$ of categories, where $\operatorname{Corr}(A,B)$ denotes the category of all $A$-$B$-correspondences, $\operatorname{Rep}(A)$ is the category of all unital normal $*$-representations of $A$ on Hilbert spaces and $\operatorname{Fun}(\operatorname{Rep}(B), \operatorname{Rep}(A))$ denotes the category of all normal $*$-functors $\operatorname{Rep}(B)\to \operatorname{Rep}(A)$. In this paper, we upgrade the von Neumann algebras $A$ and $B$ with actions $A\curvearrowleft \mathbb{G}$ and $B\curvearrowleft \mathbb{G}$ of a locally compact quantum group $\mathbb{G}$, and we provide several equivariant versions of the $W^*$-algebraic Eilenberg-Watts theorem using the language of module categories. We also prove that for a locally compact quantum group $\mathbb{G}$ with Drinfeld double $D(\mathbb{G})$, the category of unitary $D(\mathbb{G})$-representations is isomorphic to the Drinfeld center of $\operatorname{Rep}(\mathbb{G})$, generalizing a result by Neshveyev-Yamashita from the compact to the locally compact setting.