  We study the thermal equilibrium states (KMS states) of infinitely degenerate
Hamiltonians, in particular, we study the example of the Landau levels. We
classify all KMS states in an example of algebra suitable for describing
infinitely degenerate systems and we show that there is no cyclic and
separating vector corresponding to the Landau Hamiltonian. Then, we try to
reproduce the thermodynamical limit of a finite box as used in the very
beginning of the theory of KMS states by Haag Hugenholtz and Winnink. Finally,
we discuss the situation from the point of view of non-$\sigma$-additive
probabilities, non-normal nor semifinite states, singular (Dixmier) states and,
hence, an extension of the concept of KMS state.
