One of the most basic examples in noncommutative geometry is the so called noncommutative torus to be denoted by $\mathbb{T}_{\theta}$. As far as I know, there are several equivalent constructions of it: as a $C^*$algebra of the foliation, as the crossed product or as a universal $C^*$algebra. I'm interested in the last presentation. It is defined (I follow M. Khalkali's book "Basic noncommutative geometry") as a universal unital $C^*$algebra generated by two unitaries $u,v$ with relation $uv=\lambda vu$ where $\lambda=e^{2\pi i \theta}$. Author describes the concrete realisation of $\mathbb{T}_{\theta}$: he defines two unitary operators $U,V:L^2(S^1) \to L^2(S^1)$ by the formulas:
$$Uf(x)=e^{2\pi ix}f(x), Vf(x)=f(x+\theta).$$
(where we think of $S^1$ as of $\mathbb{R}/\mathbb{Z}$ to keep additive notation) and form the $C^*$algebra generated by these two unitaries. Then he omits the proof that this $C^*$algebra is indeed universal. I've asked one person which is more familiar with noncommutative geometry and this person said that this is folklore and he said that he doesn't know where I could find it in literature. I would like to know whether there is some standard procedure to handle such examples or maybe rather each example needs particular method?
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For a $C^\ast$algebraic approach to noncommutative two dimensional torus you may want to look at the Marc Rieffel paper $C^\ast$algebras associated with irrational rotations. For more detailed study of noncommutative torus including higher dimensions I suggest the book Elements of noncommutative geometry. About your question: Marc Rieffel has a deformation quantization theory based on operator algebras whose one of the examples is noncommutative torus, see his paper Deformation quantization for actions of $R^d$. Of course noncommutative torus is a very special case and admits many different interpretations as a noncommutative space, as you said! 

