Quantum Traces in Quantum Teichm\"uller Theory

We prove that for the torus with one hole and p greater than or equal to 1 punctures and the sphere with four holes there is a family of quantum trace functions in the quantum Teichm"uller space, analog to the non-quantum trace functions in Teichm"uller space, satisfying the properties proposed by Chekhov and Fock in their paper “Observables in 3D Gravity and Geodesic Algebras.”