On Cluster C⁎-Algebras

Abstract

We introduce a C⁎-algebra A(x,Q) attached to the cluster x and a quiver Q. If QT 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,QT) 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,QT).