Quantum graphs in infinite-dimensions: Hilbert--Schmidts and Hilbert modules

Abstract: We develop two approaches to Quantum (or Non-commutative) Graphs based on arbitrary von Neumann algebras $M\subseteq\mathcal B(H)$: one looking at operator bimodules of Hilbert--Schmidt (instead of bounded) operators, and the second looking at Quantum Adjacency Operators. Hilbert--Schmidt Quantum Graphs relate to Weaver's picture of Quantum Graphs in a complex way: by defining certain hull operations, we find a bijection between certain subsets of both objects. Given a nfs weight $\varphi$ on $M$ the operator-valued weight $\varphi^{-1}$ can be defined, as considered by Wasilewski for direct sums of matrix algebras. We show how to build a natural self-dual Hilbert $C^*$-module from this, which mediates a bijection between HS Quantum Relations and projections $e\in M\bar\otimes M^{\text{op}}$. When $e$ is integrable for the slice-map $\operatorname{id}\otimes\varphi^{\text{op}}$ there is a related normal CP map $A\colon M\to M$: this is a Quantum Adjacency Operator, which has a Kraus operator representation built from the HS Quantum Relation. When $e$ and its tensor swap map are both integrable, we find certain symmetries of $A$. We illustrate our theory by a careful consideration of certain examples, including detailed links with the finite-dimensional setting.