A Few Observations on Weaver's Quantum Relations

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.