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 a 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 Teichmueller 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)).Comment: to appear Journal of Function Space
