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A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations
Greg Kuperberg, University of California, Davis, CA, and Nik Weaver, Washington University, St. Louis, MO
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Memoirs of the American Mathematical Society
2011; 140 pp; softcover
Volume: 215
ISBN-10: 0-8218-5341-4
ISBN-13: 978-0-8218-5341-2
List Price: US$78 Individual Members: US$46.80
Institutional Members: US\$62.40
Order Code: MEMO/215/1010

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 $$\mathcal{M}{\overline{\otimes}} \mathcal{B}(l^2)$$.

A von Neumann Algebra Approach to Quantum Metrics by Greg Kuperberg and Nik Weaver
• Introduction
• Measurable and quantum relations
• Quantum metrics
• Examples
• Lipschitz operators
• Quantum uniformities
• Bibliography
Quantum Relations by Nik Weaver
• Introduction
• Measurable relations
• Quantum relations
• Bibliography
• Notation index
• Subject index