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A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations

Greg Kuperberg University of California, Davis, Davis, CA
Nik Weaver Washington University, St. Louis, MO
Available Formats:
Electronic ISBN: 978-0-8218-8512-3
Product Code: MEMO/215/1010.E
List Price: $78.00 MAA Member Price:$70.20
AMS Member Price: $46.80 Click above image for expanded view A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations Greg Kuperberg University of California, Davis, Davis, CA Nik Weaver Washington University, St. Louis, MO Available Formats:  Electronic ISBN: 978-0-8218-8512-3 Product Code: MEMO/215/1010.E  List Price:$78.00 MAA Member Price: $70.20 AMS Member Price:$46.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 2152011; 140 pp
MSC: Primary 46; 28; Secondary 81; 54;

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
• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2152011; 140 pp
MSC: Primary 46; 28; Secondary 81; 54;

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
• 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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