  Given a stationary state for a noncommutative flow, we study a boundedness
condition, depending on a positive parameter beta, which is weaker than the KMS
equilibrium condition at inverse temperature beta. This condition is equivalent
to a holomorphic property closely related to the one recently considered by
Ruelle and D'Antoni-Zsido and shared by a natural class of non-equilibrium
steady states. Our holomorphic property is stronger than the Ruelle's one and
thus selects a restricted class of non-equilibrium steady states. We also
introduce the complete boundedness condition and show this notion to be
equivalent to the Pusz-Woronowicz complete passivity property, hence to the KMS
condition.
  In Quantum Field Theory, the beta-boundedness condition can be interpreted as
the property that localized state vectors have energy density levels increasing
beta-subexponentially, a property which is similar in the form and weaker in
the spirit than the modular compactness-nuclearity condition. In particular,
for a Poincare' covariant net of C*-algebras on the Minkowski spacetime, the
beta-boundedness property, for beta greater equal than 2 pi, for the boosts is
shown to be equivalent to the Bisognano-Wichmann property. The Hawking
temperature is thus minimal for a thermodynamical system in the background of a
Rindler black hole within the class of beta-holomorphic states. More generally,
concerning the Killing evolution associated with a class of stationary quantum
black holes, we characterize KMS thermal equilibrium states at Hawking
temperature in terms of the boundedness property and the existence of a
translation symmetry on the horizon.
