Journal of the
Korean Mathematical Society JKMS

We study a class of KMS-symmetric quantum Markovian semigroups on a quantum spin system $(\mathcal A,\tau,\omega)$, where $\mathcal A$ is a quasi-local algebra, $\tau$ is a strongly continuous one parameter group of $*$-automorphisms of $\mathcal A$ and $\omega$ is a Gibbs state on $\mathcal A$. The semigroups can be considered as the extension of semigroups on the nontrivial abelian subalgebra. Let $\mathcal H$ be a Hilbert space corresponding to the GNS representation constructed from $\omega$. Using the general construction method of Dirichlet form developed in [8], we construct the symmetric Markovian semigroup $\{T_t\}_{t\ge0}$ on $\mathcal H$. The semigroup $\{T_t\}_{t\ge0}$ acts separately on two subspaces $\mathcal H_d$ and $\mathcal H_{od}$ of $\mathcal H$, where $\mathcal H_d$ is the diagonal subspace and $\mathcal H_{od}$ is the off-diagonal subspace, $\mathcal H=\mathcal H_d\oplus\mathcal H_{od}$. The restriction of the semigroup $\{T_t\}_{t\ge0}$ on $\mathcal H_d$ is Glauber dynamics, and for any $\eta\in\mathcal H_{od}$, $T_t\eta$ decays to zero exponentially fast as $t$ approaches to the infinity.

Keywords: KMS symmetric quantum Markovian semigroups, quantum spin systems, diagonal subspace, Glauber dynamics

MSC numbers: 46L55, 37A60