A von Neumann algebra approach to quantum metrics

@article{Kuperberg2010AVN,
  title={A von Neumann algebra approach to quantum metrics},
  author={Greg Kuperberg and Nik Weaver},
  journal={arXiv: Operator Algebras},
  year={2010}
}
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. 
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References

SHOWING 1-10 OF 22 REFERENCES
Gromov-Hausdorff Distance for Quantum Metric Spaces/Matrix Algebras Converge to the Sphere for Quantum Gromov-Hausdorff Distance
Gromov-Hausdorff distance for quantum metric spaces Bibliography Matrix algebras Converge to the sphere for quantum Gromov-Hausdorff distance Bibliography.
THE METRIC ASPECT OF NONCOMMUTATIVE GEOMETRY
Most of the previous work on “noncommutative geometry” could more accurately be labeled as noncommutative differential topology, in that it deals with the homology of differential forms on
Finite propagation speed and causal free quantum fields on networks
Laplace operators on metric graphs give rise to Klein–Gordon and wave operators. Solutions of the Klein–Gordon equation and the wave equation are studied and finite propagation speed is established.
An Introduction to Dirac Operators on Manifolds
Introduction * Clifford Algebras * Manifolds * Dirac Operators * Conformal Maps * Unique Continuation and the Cauchy Kernel * Boundary Values * Appendix. General Manifolds * The Additional Canterbury
Tensor Products of Operator Spaces II
  • D. Blecher
  • Mathematics
    Canadian Journal of Mathematics
  • 1991
Abstract Together with Vern Paulsen we were able to show that the elementary theory of tensor norms of Banach spaces carries over to operator spaces. We suggested that the Grothendieck tensor norm
Banach algebras and the general theory of *-algebras
9. *-algebras 10. Special *-algebras 11. Banach *-algebras 12. Locally compact groups and their *-algebras Bibliography Index Symbol index.
Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras
1. Numerical range in unital normed algebras 2. Hermitian elements of a complex unital Banach algebra 3. Operators 4. Some recent developments.
Continuous fields ofC*-algebras coming from group cocycles and actions
Recently I have been attempting to formulate a suitable C*-algebraic framework for the subject of deformation quantization [-3, 19]. Continuous fields of C*-algebras provide one of the key elements
Theory of quantum error correction for general noise
TLDR
This work shows that a suitable notion of "number of errors" e makes sense for any quantum or classical system in the presence of arbitrary interactions, and proves the existence of large codes for both quantum and classical information.
Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
where dEλ is the projection valued measure associated with /^Δ". A natural problem is to study the behavior of the explicit kernel kf(X)(xx, x2) representing /(/^Δ), in terms of the behavior of
...
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