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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
A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations
eBook 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
A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations
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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
eBook 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 \(\mathcal{M}{\overline{\otimes}} \mathcal{B}(l^2)\).

  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    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 publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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