Steps toward the modular functor conjecture for quantized geodesic lengths

Let $\mathfrak{S} = \Sigma \setminus \mathcal{P}$ be a triangulable marked surface, let $\Delta$ be an ideal triangulation, and let $\hbar \in \mathbb{R}_{>0}$. For an essential simple loop $\gamma$, write $\rho^\hbar_\Delta([K_\gamma])$ for its skein-algebra operator on the Schwartz space $\mathscr{S}_\Delta \subset \mathscr{H}_\Delta$, and denote its self-adjoint extension by $\mathbf{f}^\hbar_{[\gamma],\Delta}$ and the associated quantized length operator by $\mathbf{l}^\hbar_{[\gamma],\Delta}$. Steps toward the modular functor conjecture. (1) For each essential simple loop $\gamma$ in $\mathfrak{S}$, $\rho^\hbar_\Delta([K_\gamma])$ is essentially self-adjoint on $\mathscr{S}_\Delta$. (2) The unique self-adjoint extension $\mathbf{f}^\hbar_{[\gamma],\Delta}$ has simple spectrum $[2,\infty)$, and there is a unique self-adjoint operator $\mathbf{l}^\hbar_{[\gamma],\Delta}$ with simple spectrum $[0,\infty)$ satisfying the defining equation for the quantized length operator. (3) For any two disjoint essential simple loops $\gamma$ and $\xi$ in $\mathfrak{S}$, the operators $\mathbf{f}^\hbar_{[\gamma],\Delta}$ and $\mathbf{f}^\hbar_{[\xi],\Delta}$ strongly commute. These properties are proposed as initial steps toward the direct-integral decomposition and equivariance required by the modular functor conjecture of Fock and Goncharov; their resolution is not supplied in the given text.