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

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(h0) be the CCR algebra over a complex separable pre-Hilbert space h0 and let ω be a quasi-free state on A(h0). For any normalized admissible function f and complete orthonormal system (CONS) {gn}⊂h0, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(h0),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.