One of the most basic examples in non-commutative geometry is the so-called non-commutative torus, to be denoted by $ \mathbb{T}_{\theta} $. As far as I know, there are several equivalent constructions of it:

1. As the $ C^{*} $-algebra of a foliation.
2. As a crossed-product $ C^{*} $-algebra.
3. As a universal $ C^{*} $-algebra.

I’m interested in the last construction. It’s defined (I’m following M. Khalkhali’s book Basic Noncommutative Geometry) as a universal unital $ C^{*} $-algebra generated by two unitaries $ u $ and $ v $ with relation $ u v = \lambda v u $, where $ \lambda = e^{2 \pi i \theta} $. The author describes the concrete realization of $ \mathbb{T}_{\theta} $ as follows. He defines two unitary operators $ U,V: {L^{2}}(\mathbb{S}^{1}) \to {L^{2}}(\mathbb{S}^{1}) $ by the formulas $$ Uf(x) = e^{2 \pi i x} \cdot f(x), \quad Vf(x) = f(x + \theta) $$ (where we think of $ \mathbb{S}^{1} $ as $ \mathbb{R} / \mathbb{Z} $ to ensure that addition makes sense) and forms the $ C^{*} $-algebra generated by these unitaries. He then omits the proof that this $ C^{*} $-algebra is indeed universal.

I’ve asked someone who is more familiar with noncommutative geometry, and he said that this is a folklore result and that he doesn’t know where I can find it in the literature. Hence, I would like to know if there is a standard procedure to handle such examples or if each example needs a particular method?

One of the most basic examples in noncommutative geometry is the so-called noncommutative torus, denoted here by $ \mathbb{T}_{\theta} $. As far as I know, there are several equivalent constructions of it:

1. as the $ C^{*} $-algebra of a foliation;
2. as a crossed-product $ C^{*} $-algebra;
3. as a universal $ C^{*} $-algebra.

I’m interested in the last construction. It is defined (here I follow M. Khalkhali’s book Basic Noncommutative Geometry) as a universal unital $ C^{*} $-algebra generated by two unitaries $ u $ and $ v $ with relation $ u v = \lambda v u $, where $ \lambda = e^{2 \pi i \theta} $. The author describes the concrete realisation of $ \mathbb{T}_{\theta} $ as follows. He defines two unitary operators $ U,V: {L^{2}}(\mathbb{S}^{1}) \to {L^{2}}(\mathbb{S}^{1}) $ by the formulas $$ Uf(x) = e^{2 \pi i x} \cdot f(x), \quad Vf(x) = f(x + \theta) $$ (where we think of $ \mathbb{S}^{1} $ as $ \mathbb{R} / \mathbb{Z} $ to keep additive notation) and forms the $C^{*} $-algebra generated by these two unitaries. Then he then omits the proof that this $ C^{*} $-algebra is indeed universal.

I’ve asked one person who is more familiar than me with noncommutative geometry, and this person said that this is folklore and that he doesn’t know where I could find it in the literature. Hence, I would like to know if there is some standard procedure to handle such examples or if each example needs a particular method?

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?

One of the most basic examples in non-commutative geometry is the so-called non-commutative torus, to be denoted by $ \mathbb{T}_{\theta} $. As far as I know, there are several equivalent constructions of it:

1. As the $ C^{*} $-algebra of a foliation.
2. As a crossed-product $ C^{*} $-algebra.
3. As a universal $ C^{*} $-algebra.

I’m interested in the last construction. It’s defined (I’m following M. Khalkhali’s book Basic Noncommutative Geometry) as a universal unital $ C^{*} $-algebra generated by two unitaries $ u $ and $ v $ with relation $ u v = \lambda v u $, where $ \lambda = e^{2 \pi i \theta} $. The author describes the concrete realization of $ \mathbb{T}_{\theta} $ as follows. He defines two unitary operators $ U,V: {L^{2}}(\mathbb{S}^{1}) \to {L^{2}}(\mathbb{S}^{1}) $ by the formulas $$ Uf(x) = e^{2 \pi i x} \cdot f(x), \quad Vf(x) = f(x + \theta) $$ (where we think of $ \mathbb{S}^{1} $ as $ \mathbb{R} / \mathbb{Z} $ to ensure that addition makes sense) and forms the $ C^{*} $-algebra generated by these unitaries. He then omits the proof that this $ C^{*} $-algebra is indeed universal.

I’ve asked someone who is more familiar with noncommutative geometry, and he said that this is a folklore result and that he doesn’t know where I can find it in the literature. Hence, I would like to know if there is a standard procedure to handle such examples or if each example needs a particular method?