On Cluster C ⁎ -Algebras

We introduce a C ⁎ -algebra A ( x , Q ) attached to the cluster x and a quiver Q . If Q T is the quiver coming from triangulation T of the Riemann surface S with a finite number of cusps, we prove that the primitive spectrum of A ( x , Q T ) times R is homeomorphic to a generic subset of the Teichmüller space of surface S . We conclude with an analog of the Tomita-Takesaki theory and the Connes invariant T ( M ) for the algebra A ( x , Q T ) .