### Abstract

Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1-14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(Heng hooktop sign_{0}) be the CCR algebra over a complex separable pre-Hilbert space Heng hooktop sign_{0} and let ω be a quasi-free state on A(Heng hooktop sign_{0}). For any normalized admissible function f and complete orthonormal system (CONS) {g_{n}}Heng hooktop sign_{0}, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(Heng hooktop sign_{0}), ω). It turns out that the form is independent of admissible function f and CONS {g_{n}} chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

Original language | English |
---|---|

Pages (from-to) | 723-753 |

Number of pages | 31 |

Journal | Journal of Mathematical Physics |

Volume | 44 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2003 Feb 1 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*44*(2), 723-753. https://doi.org/10.1063/1.1532770

}

*Journal of Mathematical Physics*, vol. 44, no. 2, pp. 723-753. https://doi.org/10.1063/1.1532770

**Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasi-free states.** / Bahn, Changsoo; Ko, Chul Ki; Park, Yong Moon.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasi-free states

AU - Bahn, Changsoo

AU - Ko, Chul Ki

AU - Park, Yong Moon

PY - 2003/2/1

Y1 - 2003/2/1

N2 - Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1-14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(Heng hooktop sign0) be the CCR algebra over a complex separable pre-Hilbert space Heng hooktop sign0 and let ω be a quasi-free state on A(Heng hooktop sign0). For any normalized admissible function f and complete orthonormal system (CONS) {gn}Heng hooktop sign0, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(Heng hooktop sign0), ω). It turns out that the form is independent of admissible function f and CONS {gn} chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

AB - Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1-14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(Heng hooktop sign0) be the CCR algebra over a complex separable pre-Hilbert space Heng hooktop sign0 and let ω be a quasi-free state on A(Heng hooktop sign0). For any normalized admissible function f and complete orthonormal system (CONS) {gn}Heng hooktop sign0, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(Heng hooktop sign0), ω). It turns out that the form is independent of admissible function f and CONS {gn} chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

UR - http://www.scopus.com/inward/record.url?scp=0037326790&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037326790&partnerID=8YFLogxK

U2 - 10.1063/1.1532770

DO - 10.1063/1.1532770

M3 - Article

AN - SCOPUS:0037326790

VL - 44

SP - 723

EP - 753

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 2

ER -