KMS states on the C*-algebras of non-principal groupoids
  We describe KMS-states on the C*-algebras of etale groupoids in terms of measurable fields of traces on the C*-algebras of the isotropy groups. We use this description to analyze tracial states on the transformation groupoid C*-algebras and to give a short proof of recent results of Cuntz, Deninger and Laca on the Toeplitz algebras of the ax+b semigroups of the rings of integers in number fields. 
Sergey Neshveyev
math.OA