Quantization of of an elliptic curve can be done in different ways. In C^*-algebraic version, one can start with the C^*-algebra of continuous functions on ordinary torus and by inserting a deformation parameter \theta into the product obtain a deformed non-commutaive C^*-algebra of functions on the quantum torus.

My question is:

Is there any natural way for deformation quantization of closed Riemann surfaces with higher genus in the above sense?