The concept of quantum relation $\mathcal{ R}$ over a von Neumann algebra $\mathcal{ M}$ has been recently introduced by Nik Weaver. When $\mathcal{ M}$ is either finite dimensional or discrete and abelian, $\mathcal{ R}$ is given by an orthogonal projection in $\mathcal{ M} \otimes \mathcal{ M}_\mathrm{ op}$. Here, we generalize such result to general von Neumann algebras, proving that quantum relations are in bijective correspondence with weak-$\ast$ closed left ideals inside $\mathcal{ M} \otimes_{ e h} \mathcal{ M}$, where $\otimes_{ e h}$ is the extended Haagerup tensor product. The correspondence between the two is given by identifying $\mathcal{ M} \otimes_{ e h} \mathcal{ M}$ with $\mathcal{ M}'$-bimodular operators and proving a double annihilator relation. Given an action of a group/quantum group on $\mathcal{ M}$ we give a definition for invariant quantum relations and prove that, in the case of group von Neumann algebras $\mathcal{ L} G$, invariant quantum relations are left ideals in the measure algebra $M G$. At the end we explore possible applications to noncommutative harmonic analysis, in particular noncommutative Gaussian bounds.
Categories: math.OA math.FA