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In PT-quantum mechanics the generator of the dynamics of a physical system isnot necessarily a self-adjoint Hamiltonian. It is now clear that this choicedoes not prevent to get a unitary time evolution and a real spectrum of theHamiltonian, even if, most of the times, one is forced to deal withbiorthogonal sets rather than with on orthonormal basis of eigenvectors. Inthis paper we consider some extended versions of the Heisenberg algebraicdynamics and we relate this analysis to some generalized version of Gibbsstates and to their related KMS-like conditions. We also discuss somepreliminary aspects of the Tomita-Takesaki theory in our context.

Gibbs states, algebraic dynamics and generalized Riesz systems