### Abstract

We study a class of KMS-symmetric quantum Markovian semigroups on a quantum spin system (A, τ, ω), where A is a quasi-local algebra, τ is a strongly continuous one parameter group of *-automorphisms of A and ω is a Gibbs state on A. The semigroups can be considered as the extension of semigroups on the nontrivial abelian subalgebra. Let H be a Hubert space corresponding to the GNS representation constructed from ω. Using the general construction method of Dirichlet form developed in [8], we construct the symmetric Markovian semigroup {T_{t}}_{t≥0} on H.The semigroup {T_{t}}_{t≥0} acts separately on two subspaces Hd and H_{0d} of H where H_{d} is the diagonal subspace and H _{od} is the off-diagonal subspace, H= H_{d} ⊕ H _{od}. The restriction of the semigroup {Tt}_{t≥0} on H _{d} is Glauber dynamics, and for any η € H_{0d}, T_{t}η decays to zero exponentially fast as t approaches to the infinity.

Original language | English |
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Pages (from-to) | 1075-1087 |

Number of pages | 13 |

Journal | Journal of the Korean Mathematical Society |

Volume | 45 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2008 Jul |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Journal of the Korean Mathematical Society*,

*45*(4), 1075-1087. https://doi.org/10.4134/JKMS.2008.45.4.1075