A few observations on Weaver's quantum relations

Abstract

Recently, a notion of quantum relation over a von Neumann algebra M has been introduced by Weaver. That definition generalizes the concept of a relation over a set. We prove that quantum relations over M are in bijective correspondence with weakly closed left ideals in M⊗ehM, where ⊗eh 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.