A Few Observations on Weaver's Quantum Relations

TL;DR

This paper extends the theory of quantum relations over von Neumann algebras, establishing a bijective correspondence with certain ideals in the extended Haagerup tensor product, and explores invariance under group actions with applications to noncommutative harmonic analysis.

Contribution

It generalizes the characterization of quantum relations to all von Neumann algebras using the extended Haagerup tensor product and introduces invariance concepts under group actions.

Findings

Quantum relations correspond to weak-* closed left ideals in the extended Haagerup tensor product.

Invariant quantum relations in group von Neumann algebras are characterized as left ideals in the measure algebra.

Potential applications to noncommutative harmonic analysis and Gaussian bounds.

Abstract

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.