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