I find myself needing to construct some noncommutative geometries. I want to take various (algeba-) geometric objects and look at their noncommutative analogs. Is there a constructive way to do this? It seems hard to find a topological space's algebra of continuous functions. I do see examples of $\mathbb{C}^n$ and $\mathbb{P}^n$ and how to proceed if you have a variety. But even for $\mathbb{P}^2$ or $\mathbb{P}^3$, I have not been able to find WHY those* algebras are appropriate. I have also found the noncommutative torus, but I can't find any explanation as how to derive this from the torus. So to sum up, I guess my question is if I have an arbitrary geometric object, say a complex manifold, then is there any general way to create a noncommutative analog?

*$\mathbb{P}^3 \cong T(z_1, z_2, z_3,z_4)/ \langle[z_3, z_i] = [z_4, z_i] = 0, [z_1, z_2] = 2hz_3z_4\rangle$ where $T$ is the tensor algebra.

Why do we make $z_3$ and $z_4$ commute with everything? Why is the commutator of $z_1$ and $z_2$ proportional to $z_3z_4$ and not, say, $z_3^2$?