It is shown that in the regular viz. phase space representation of a Moyal quantization real polynomials in the phase space variables become essentially self-adjoint operators, and that functions in the positive cone of monomials of even power in the phase space variables yield positive operators. Conditions on the boundedness of Moyal operators are given. The classical limit of a Moyal operator is defined and used to relate classical with quantum entropy.

Abstract. For the concept of intrinsic stochasticity as introduced by Prigogine et al., a general mathematical approach is outlined. It uses W*-algebras ~¢ with a trace z of dynamical observables, identifying the state space with H = L2(~, z). The main result is that the incorporation of Lyapunov processes in H leads necessarily to the larger algebra LP(H). This induces a strictly ascending chain of algebras of observables of increasing complexity. AMS (MOS) subject classifications. 93D05, 93D25, 82A15, 82A05, 82A35.

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We give first an approximation of the operator δ h : f → δ h f := h * h f −f * h h in terms of h2n , n 0, where h ≡ h(p, q), (p, q) ∈ R 2n , is a Hamilton function and * h denotes the star product. The operator, which is the generator of time translations in a * h-algebra, can be considered as a canonical extension of the Liouville operatorUsing this operator we investigate the dynamics and trajectories of some examples with a scheme that extends the Hamilton-Jacobi method for classical dynamics to Moyal dynamics. The examples we have chosen are Hamiltonians with a one-dimensional quartic potential and two-dimensional radially symmetric nonrelativistic and relativistic Coulomb potentials, and the Hamiltonian for a Schwarzschild metric. We further state a conjecture concerning an extension of the Bohr-Sommerfeld formula for the calculation of the exact eigenvalues for systems with classically periodic trajectories.