  D = 2+1 gravity with a cosmological constant has been shown by Bonzom and
Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. We
make use of this fact to show that, for positive cosmological constant, the
Lorentzian theory can be partially gauge fixed and reduced to an SU(2)
Chern-Simons theory. We then review the already known quantization of the
latter in the framework of Loop Quantization for the case of space being
topogically a cylinder. We finally construct, in the same setting, a quantum
observable which, although non-trivial at the quantum level, corresponds to a
null classical quantity.
