  We introduce a non-commutative generalization of the notion of (approximately
proper) equivalence relation and propose the construction of a "quotient
space". We then consider certain one-parameter groups of automorphisms of the
resulting C*-algebra and prove the existence of KMS states at every
temperature. In a model originating from Thermodynamics we prove that these
states are unique as well. We also show a relationship between maximizing
measures (the analogue of the Aubry-Mather measures for expanding maps) and
ground states. In the last section we explore an interesting example of phase
transitions.
