# Realisation of noncommutative torus

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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Related: en.wikipedia.org/wiki/… –  Qiaochu Yuan Feb 12 at 1:32
Ken Davidson's C*-algebra book has a proof of this fact –  Caleb Eckhardt Feb 12 at 1:54
Thank You, I've checked in his book (You probably mean $C^*$-algebras by example) but I'm a bit confused because his construction is different: he uses $U$ and $V$ as above only for ensuring that there is at least one $C^*$-algebra with two unitaries satisfying such a relation. But he constructs universal $C^*$-algebra by taking all the irreducible pairs $(U_i,V_i)$ satisfying this relation. Then he formes $U=\bigoplus_{i}U_i, V=\bigoplus_{i}V_i$ so at the moment I'm not sure: noncommutative torus acts on the direct sum $\bigoplus_{i}L^2(S^1)_i$? –  truebaran Feb 12 at 20:21
@truebaran In Theorem VI.1.4 he shows that the universal C*-algebra generated by two unitaries $u,v$ that satisfy the relation $uvu^*v^*=\lambda$ is simple. From this one deduces that whenever two unitaries satisfy $uvu^*v^*=\lambda$, they must generate a copy of $A_\theta$ –  Caleb Eckhardt Feb 12 at 21:30

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!