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Quantum stochastic convolution cocycles II

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Quantum stochastic convolution cocycles II. / Lindsay, J. Martin; Skalski, Adam G.
In: Communications in Mathematical Physics, Vol. 280, No. 3, 06.2008, p. 575-610.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Lindsay, JM & Skalski, AG 2008, 'Quantum stochastic convolution cocycles II', Communications in Mathematical Physics, vol. 280, no. 3, pp. 575-610. https://doi.org/10.1007/s00220-008-0465-x

APA

Lindsay, J. M., & Skalski, A. G. (2008). Quantum stochastic convolution cocycles II. Communications in Mathematical Physics, 280(3), 575-610. https://doi.org/10.1007/s00220-008-0465-x

Vancouver

Lindsay JM, Skalski AG. Quantum stochastic convolution cocycles II. Communications in Mathematical Physics. 2008 Jun;280(3):575-610. doi: 10.1007/s00220-008-0465-x

Author

Lindsay, J. Martin ; Skalski, Adam G. / Quantum stochastic convolution cocycles II. In: Communications in Mathematical Physics. 2008 ; Vol. 280, No. 3. pp. 575-610.

Bibtex

@article{4ccff6bfe63840bf9bd1d92985d843b1,
title = "Quantum stochastic convolution cocycles II",
abstract = "Schurmann's theory of quantum Levy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*-hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *-homomorphic convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum Levy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Levy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups. ",
keywords = "Noncommutative probability, quantum stochastic , compact quantum group , C*-bialgebra , C*-hyperbialgebra , operator space , stochastic cocycle , quantum Levy process",
author = "Lindsay, {J. Martin} and Skalski, {Adam G.}",
note = "The original publication is available at www.link.springer.com",
year = "2008",
month = jun,
doi = "10.1007/s00220-008-0465-x",
language = "English",
volume = "280",
pages = "575--610",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "3",

}

RIS

TY - JOUR

T1 - Quantum stochastic convolution cocycles II

AU - Lindsay, J. Martin

AU - Skalski, Adam G.

N1 - The original publication is available at www.link.springer.com

PY - 2008/6

Y1 - 2008/6

N2 - Schurmann's theory of quantum Levy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*-hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *-homomorphic convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum Levy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Levy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups.

AB - Schurmann's theory of quantum Levy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*-hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *-homomorphic convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum Levy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Levy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups.

KW - Noncommutative probability

KW - quantum stochastic

KW - compact quantum group

KW - C-bialgebra

KW - C-hyperbialgebra

KW - operator space

KW - stochastic cocycle

KW - quantum Levy process

U2 - 10.1007/s00220-008-0465-x

DO - 10.1007/s00220-008-0465-x

M3 - Journal article

VL - 280

SP - 575

EP - 610

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -