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 Neu-mann algebras developed in Pa], we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(h 0) be the CCR algebra over a complex separable pre-Hilbert space h 0 and let ! be a quasi-free state on A(h 0). For any normalized admissible function f and complete orthonormal system (CONS) fg n g h 0 , we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(h 0); !). It turns out that the form is independent of admissible function f and CONS fg n g 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.