On cluster C*-algebras

TL;DR

This paper introduces a new class of C*-algebras associated with cluster structures and quivers, establishing a topological correspondence with Teichmüller spaces and exploring their modular theory properties.

Contribution

It constructs C*-algebras from cluster data and proves their primitive spectrum relates to Teichmüller space, extending operator algebra theory to geometric structures.

Findings

Primitive spectrum homeomorphic to a subset of Teichmüller space

Established an analog of Tomita-Takesaki theory for these algebras

Identified the Connes invariant for the constructed C*-algebras

Abstract

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)).