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

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) {g_n}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 {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.