Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras
We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the...
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| Vydané v: | Communications in mathematical physics Ročník 303; číslo 2; s. 555 - 594 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Berlin/Heidelberg
Springer-Verlag
01.04.2011
Springer |
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| ISSN: | 0010-3616, 1432-0916 |
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| Abstract | We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on
for all
n
≥ 3, as well as an example of a one-parameter semigroup (
T
(
t
))
t
≥0
of Markov maps on
such that
T
(
t
) fails to be factorizable for all small values of
t
> 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality. |
|---|---|
| AbstractList | We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on
for all
n
≥ 3, as well as an example of a one-parameter semigroup (
T
(
t
))
t
≥0
of Markov maps on
such that
T
(
t
) fails to be factorizable for all small values of
t
> 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality. |
| Author | Haagerup, Uffe Musat, Magdalena |
| Author_xml | – sequence: 1 givenname: Uffe surname: Haagerup fullname: Haagerup, Uffe organization: Department of Mathematical Sciences, University of Copenhagen – sequence: 2 givenname: Magdalena surname: Musat fullname: Musat, Magdalena email: musat@math.ku.dk organization: Department of Mathematical Sciences, University of Copenhagen |
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| Cites_doi | 10.1007/s00222-008-0137-7 10.1016/0024-3795(93)90274-R 10.1103/PhysRevA.72.052317 10.1007/s00220-009-0824-2 10.1007/BF01232444 10.1016/0022-1236(82)90022-2 10.1016/0024-3795(75)90075-0 10.1007/BF01420337 10.1007/BF01205670 10.1353/ajm.0.0122 10.1016/0022-1236(72)90004-3 10.1007/978-1-4612-1128-0 10.1112/plms/s3-7.1.29 10.1016/0022-1236(85)90084-9 10.1007/s00222-002-0235-x 10.1090/S0002-9939-08-09452-5 10.1142/S0129167X04002417 10.1007/s00440-005-0456-1 10.1016/j.jfa.2008.09.021 10.1007/978-3-662-09089-3 10.1090/S0002-9904-1962-10737-X 10.2140/pjm.1989.137.265 10.1007/BFb0074482 10.1007/978-3-662-10451-4 10.1090/memo/0585 |
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| Issue | 2 |
| Keywords | Embedding Problem Asymptotic Quantum Normal Faithful State Noncommutative Setting Tracial State Lower bound Semigroup Quantum information Asymptotic approximation Mathematical physics Quantum theory Information theory |
| Language | English |
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| SubjectTerms | Classical and Quantum Gravitation Complex Systems Exact sciences and technology Functional analysis Group theory Group theory and generalizations Mathematical analysis Mathematical and Computational Physics Mathematical methods in physics Mathematical Physics Mathematics Operator theory Other topics in mathematical methods in physics Physics Physics and Astronomy Quantum Physics Relativity Theory Sciences and techniques of general use Theoretical |
| Title | Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras |
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