# A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations

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*Greg Kuperberg; Nik Weaver*

In A von Neumann Algebra Approach to Quantum
Metrics, Kuperberg and Weaver propose a new definition of quantum
metric spaces, or W*-metric spaces, in the setting of von Neumann
algebras. Their definition effectively reduces to the classical notion
in the atomic abelian case, has both concrete and intrinsic
characterizations, and admits a wide variety of tractable examples. A
natural application and motivation of their theory is a mutual
generalization of the standard models of classical and quantum error
correction.

In Quantum Relations Weaver defines a “quantum
relation” on a von Neumann algebra
\(\mathcal{M}\subseteq\mathcal{B}(H)\) to be a weak* closed
operator bimodule over its commutant \(\mathcal{M}'\).
Although this definition is framed in terms of a particular
representation of \(\mathcal{M}\), it is effectively
representation independent. Quantum relations on
\(l^\infty(X)\) exactly correspond to subsets of
\(X^2\), i.e., relations on \(X\). There is also a good
definition of a “measurable relation” on a measure space,
to which quantum relations partially reduce in the general abelian
case.

By analogy with the classical setting, Weaver can identify structures
such as quantum equivalence relations, quantum partial orders, and
quantum graphs, and he can generalize Arveson's fundamental work on
weak* closed operator algebras containing a masa to these cases. He
is also able to intrinsically characterize the quantum relations on
\(\mathcal{M}\) in terms of families of projections in
\(\mathcal{M}{\overline{\otimes}} \mathcal{B}(l^2)\).

#### Table of Contents

# Table of Contents

## A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations

- A von Neumann Algebra Approach to Quantum Metrics by Greg Kuperberg and Nik Weaver 18 free
- Quantum Relations by Nik Weaver 8188