# Journal of Operator Theory

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

A few observations on Weaver's quantum relations

**Authors**:
Adri\'an M. Gonz\'alez-P\'erez

**Author institution:**LBBP, Univ. of Clermont Auvergne,\break
3 place Vasarely
Aubi{\`e}re Cedex, 63178, France

**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.

**DOI: **http://dx.doi.org/10.7900/jot.2018sep20.2249

**Keywords: **operator spaces, von Neumann algebras, bimodules, tensor products, noncommutative geometry

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