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On cluster \(C^\ast\)-algebras. (English) Zbl 1358.46055

Summary: We introduce a \(C^\ast\)-algebra \(\mathbb{A}(\mathbf{x}, Q)\) attached to the cluster \(\mathbf{x}\) 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 \(\mathbb{A}(\mathbf{x}, Q_T)\) times \(\mathbb{R}\) 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 \(T(\mathcal{M})\) for the algebra \(\mathbb{A}(\mathbf{x}, Q_T)\).

MSC:

46L05 General theory of \(C^*\)-algebras
46L40 Automorphisms of selfadjoint operator algebras
30F60 Teichmüller theory for Riemann surfaces
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