Quantum Traces in Quantum Teichmüller Theory

We prove that for the torus with one hole and finitely many punctures and the sphere with four holes there is a family of quantum trace functions in the quantum Teichmüller space, analog to the non-quantum trace functions in Teichmüller space, satisfying the properties proposed by Chekhov and Fock in their paper "Observables in 3D Gravity and Geodesic Algebras."