# Journal of Operator Theory

Volume 83, Issue 1, Winter 2020  pp. 197-228.

A few observations on Weaver's quantum relations

Summary: Recently, a notion of quantum relation over a von Neumann algebra $\mathcal{M}$ has been introduced by Weaver. That definition generalizes the concept of a relation over a set. We prove that quantum relations over $\mathcal{M}$ are in bijective correspondence with weakly closed left ideals in $\mathcal{M} \otimes_\mathrm{e h} \mathcal{M}$, where $\otimes_\mathrm{e h}$ represents the extended Haagerup tensor product. The key step of the proof is showing a double annihilator relation between operator bimodules and the bimodular maps annihilating them. As an application, we study invariant quantum relations over a group von Neumann algebra.