Equilibrium and ground states from Cayley graphs

We present a general framework for the study of KMS states of generalized gauge actions on the C⁎-algebra of a Cayley graph which is pointed by considering the neutral element of the group as a distinguished vertex. We use this framework to give a concrete description of a particular kind of KMS states on the C⁎-algebras that we call abelian KMS states. If the group is nilpotent all KMS states are abelian and our analysis gives the full picture in this case. We then describe the ground states that are limits of abelian KMS states when the temperature goes to zero.