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A von Neumann Algebra Approach to Quantum Metrics

About this Title

Greg Kuperberg, Department of Mathematics, University of California, Davis, California 95616 and Nik Weaver, Department of Mathematics, Washington University, St. Louis, Missouri 63130

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 215, Number 1010
ISBNs: 978-0-8218-5341-2 (print); 978-0-8218-8512-3 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00637-4
Published electronically: March 29, 2011
Keywords: [A von Neumann Algebra Approach to Quantum Metrics] Quantum error correction, quantum metrics, quantum tori, spectral triples, von Neumann algebras; [Quantum Relations] Measurable metrics, measurable relations, operator reflexivity, quantum relations, quantum tori
MSC: Primary 46L89, 28A99; Secondary 46L10, 54E35, 81P70

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Table of Contents

A von Neumann Algebra Approach to Quantum Metrics by Greg Kuperberg and Nik Weaver

• Introduction
• 1. Measurable and quantum relations
• 2. Quantum metrics
• 3. Examples
• 4. Lipschitz operators
• 5. Quantum uniformities

Quantum Relations by Nik Weaver

• Introduction
• 6. Measurable relations
• 7. Quantum relations

Abstract

A von Neumann Algebra Approach to Quantum Metrics by Greg Kuperberg and Nik Weaver

We propose a new definition of quantum metric spaces, or W*-metric spaces, in the setting of von Neumann algebras. Our 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 our theory is a mutual generalization of the standard models of classical and quantum error correction.

Quantum Relations by Nik Weaver

We define 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, we can identify structures such as quantum equivalence relations, quantum partial orders, and quantum graphs, and we can generalize Arveson’s fundamental work on weak* closed operator algebras containing a masa to these cases. We are also able to intrinsically characterize the quantum relations on ${\mathcal M}$ in terms of families of projections in $\mathcal {M}\bar {\otimes } \mathcal {B}(l^2)$.