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?



Realisation of the noncommutative torus as a universal $ C^{*} $-algebra
According to what I have seen in the literature so far, the standard procedure consists of two main steps:


* Prove the existence of a universal $ C^{\*} $-algebra $ A\_{\theta} $ generated by two unitaries $ u $ and $ v $ that satisfy
$$
u v = e^{2 \pi i \theta} v u.
$$
**Note:** We are assuming that $ \theta $ is irrational.
* Prove that $ A\_{\theta} $ is simple, and conclude that the concrete realization given above is indeed universal.




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To accomplish Step 1, there are several methods. I understand that you have read Davidson’s book, so let me describe an approach different from his that is more algebraic in nature.


1. Let $ \mathcal{P} $ denote the free associative unital $ \mathbb{C} $-algebra in four indeterminates, $ u $, $ u^{\*} $, $ v $ and $ v^{\*} $, where the identity element of $ \mathcal{P} $ is denoted by $ \mathbf{1} $.
2. Let $ \mathcal{I}\_{\theta} $ denote the (not *a priori* proper) two-sided ideal
$$
\left\langle
u u^{\*} - \mathbf{1},
u^{\*} u - \mathbf{1},
v v^{\*} - \mathbf{1},
v^{\*} v - \mathbf{1},
u v - e^{2 \pi i \theta} v u
\right\rangle.
$$
3. Form the quotient $ \mathbb{C} $-algebra $ \mathcal{A}\_{\theta} \stackrel{\text{df}}{=} \mathcal{P} / \mathcal{I}\_{\theta} $.
4. Let $ \dot{\mathbf{1}} $, $ \dot{u} $, $ \dot{u}^{\*} $, $ \dot{v} $ and $ \dot{v}^{\*} $ denote the images of $ \mathbf{1} $, $ u $, $ u^{\*} $, $ v $ and $ v^{\*} $ in $ \mathcal{A}\_{\theta} $ respectively.
5. Then the following monomials are distinct in $ \mathcal{A}\_{\theta} $ and define a Hamel basis for it:
$$
\dot{\mathbf{1}}, \quad
\dot{u}^{m} \dot{v}^{n}, \quad
\dot{u}^{m} (\dot{v}^{\*})^{n}, \quad
(\dot{u}^{\*})^{m} \dot{v}^{n}, \quad
(\dot{u}^{\*})^{m} (\dot{v}^{\*})^{n}; \qquad
(m,n) \in \mathbb{N}\_{0}^{2} \setminus \{ (0,0) \}.
$$
As such, $ \mathcal{I}\_{\theta} $ is a proper ideal of $ \mathcal{P} $, and so $ \mathcal{A}\_{\theta} $ is a unital $ \mathbb{C} $-algebra.



> 
> **Comment:** Wegge-Olsen claims that Point (5) is obvious, but I beg to differ as its validity depends on a non-trivial algebraic result called the **Diamond Lemma for Ring Theory**. Davidson appears to avoid all forms of algebraic machinery by resorting to the GNS Construction. However, as a staunch believer in the Principle of Conservation of Difficulty, I think that Davidson is simply transferring all technical issues from the Diamond Lemma to the GNS Construction, which, as most operator algebraists would agree, is a highly non-trivial result in the representation theory of $ C^{\*} $-algebras.
> 
> 
> 


6. Define a $ C^{\*} $-representation of $ \mathcal{A}\_{\theta} $ to be a triple $ (A,s,t) $, where:


	* $ A $ is a unital $ C^{\*} $-algebra.
	* $ s $ and $ t $ are unitary elements of $ A $ satisfying $ s t = e^{2 \pi i \theta} t s $.
7. Given a $ C^{\*} $-representation $ (A,s,t) $ of $ \mathcal{A}\_{\theta} $, there exists a **unique** unital $ \mathbb{C} $-algebra homomorphism from $ \mathcal{P} $ to $ A $ defined by
$$
u \mapsto s, \quad
u^{\*} \mapsto s^{\*}, \quad
v \mapsto t \quad \text{and} \quad
v^{\*} \mapsto t^{\*}.
$$
Then as the homomorphism kills $ \mathcal{I}\_{\theta} $, we obtain a unital $ \mathbb{C} $-algebra homomorphism $ \pi\_{A,s,t}: \mathcal{A}\_{\theta} \to A $, once again **unique**, that satisfies:


	* $ {\pi\_{A,s,t}}(\dot{u}) = s $ and $ {\pi\_{A,s,t}}(\dot{u}^{\*}) = s^{\*} $.
	* $ {\pi\_{A,s,t}}(\dot{v}) = t $ and $ {\pi\_{A,s,t}}(\dot{v}^{\*}) = t^{\*} $.
	* $ {\pi\_{A,s,t}}(\dot{u}) ~ {\pi\_{A,s,t}}(\dot{v}) = e^{2 \pi i \theta} ~ {\pi\_{A,s,t}}(\dot{v}) ~ {\pi\_{A,s,t}}(\dot{u}) $.



> 
> **Question:** Do $ C^{\*} $-representations of $ \mathcal{A}\_{\theta} $ exist?
> 
> 
> **Answer:** Yes! The concrete realization $ (\mathscr{B}({L^{2}}(\mathbb{T})),U,V) $ described by the OP is one. Amusingly, the very existence of this concrete realization shows that $ \mathcal{I}\_{\theta} $ is a proper two-sided ideal of $ \mathcal{P} $, but I think we can safely say that this is not a demonstration of the fact from first principles.
> 
> 
> 


8. Define a mapping $ \| \cdot \|\_{0}: \mathcal{A}\_{\theta} \to [0,\infty] $ by
$$
\| a \|\_{0} \stackrel{\text{df}}{=}
\sup
(\{
\| {\pi\_{A,s,t}}(a) \|\_{A} \in \mathbb{R}\_{\geq 0} \mid
\text{$ (A,s,t) $ is a $ C^{\*} $-representation of $ \mathcal{A}\_{\theta} $}
\})
$$
for each $ a \in \mathcal{A}\_{\theta} $.
9. Proof sketch that $ \| \cdot \|\_{0} $ is a $ \mathbb{C} $-algebra semi-norm:


	* Thanks to the existence of a $ C^{\*} $-representation of $ \mathcal{A}\_{\theta} $, we have
	$$
	 \left\| \dot{\mathbf{1}} \right\|\_{0}
	= \| \dot{u} \|\_{0}
	= \| \dot{u}^{\*} \|\_{0}
	= \| \dot{v} \|\_{0}
	= \| \dot{v}^{\*} \|\_{0}
	= 1.
	$$
	* As $ \dot{\mathbf{1}} $, $ \dot{u} $, $ \dot{u}^{\*} $, $ \dot{v} $ and $ \dot{v}^{\*} $ generate $ \mathcal{A}\_{\theta} $, it follows that $ \| a \|\_{0} < \infty $ for each $ a \in \mathcal{A}\_{\theta} $.
	* Knowing now that $ \| \cdot \|\_{0}: \mathcal{A}\_{\theta} \to [0,\infty) $, it is easily shown to satisfy the axioms of a $ \mathbb{C} $-algebra semi-norm.
10. Notice that $ \| \cdot \|\_{0} $ also satisfies the $ C^{\*} $-identity.
11. Let $ \mathcal{N} \stackrel{\text{df}}{=} \{ a \in \mathcal{A}\_{\theta} \mid \| a \|\_{0} = 0 \} $. Then $ \mathcal{N} $ is a $ \mathbb{C} $-subalgebra of $ \mathcal{A}\_{\theta} $.
12. Form the quotient $ \mathbb{C} $-algebra $ \mathcal{A}\_{\theta} / \mathcal{N} $ to get a pre-$ C^{\*} $-algebra, denoting the quotient norm by $ \| \cdot \| $.
13. Complete $ \mathcal{A}\_{\theta} / \mathcal{N} $ with respect to $ \| \cdot \| $ to obtain the irrational rotation $ C^{\*} $-algebra $ A\_{\theta} $.
14. Clearly, $ A\_{\theta} $ is unital and is generated by the pair $ ([\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}) $ of unitary elements. Furthermore,
$$
 [\dot{u}]\_{\mathcal{N}} [\dot{v}]\_{\mathcal{N}}
= e^{2 \pi i \theta} [\dot{v}]\_{\mathcal{N}} [\dot{u}]\_{\mathcal{N}}.
$$


This completes the construction.




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> 
> **Claim:** $ (A\_{\theta},[\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}) $ is a universal $ C^{\*} $-representation of $ \mathcal{A}\_{\theta} $.
> 
> 
> 


*Proof of Claim*


Let $ (A,s,t) $ be a $ C^{\*} $-representation of $ \mathcal{A}\_{\theta} $. Then by the definition of $ \| \cdot \|\_{0} $,
$$
\forall a \in \mathcal{A}\_{\theta}: \quad
\| {\pi\_{A,s,t}}(a) \|\_{A} \leq \| a \|\_{0} = \| [a]\_{\mathcal{N}} \|.
$$
We thus have a **unique** unital $ \* $-homomorphism $ \pi\_{A,s,t}^{\mathcal{N}}: \mathcal{A}\_{\theta} / \mathcal{N} \to A $ satisfying
$$
{\pi\_{A,s,t}^{\mathcal{N}}}([\dot{u}]\_{\mathcal{N}}) = s \quad \text{and} \quad
{\pi\_{A,s,t}^{\mathcal{N}}}([\dot{v}]\_{\mathcal{N}}) = t,
$$
and we can extend this, using continuity, to a **unique** unital $ \* $-homomorphism $ \Pi\_{A,s,t}: A\_{\theta} \to A $. In other words, $ \Pi\_{A,s,t} $ is the **only** unital $ \* $-homomorphism from $ A\_{\theta} $ to $ A $ that maps $ [\dot{u}]\_{\mathcal{N}} $ to $ s $ and $ [\dot{v}]\_{\mathcal{N}} $ to $ t $.


Now, suppose that there is another unital $ C^{\*} $-algebra $ B $ generated by two unitaries $ u' $ and $ v' $ satisfying
$$
u' v' = e^{2 \pi i \theta} v' u'
$$
such that for any $ C^{\*} $-representation $ (A,s,t) $, there exists a **unique** $ \* $-homomorphism $ \Phi\_{A,s,t}: B \to A $ that maps $ u' $ to $ s $ and $ v' $ to $ t $.


The following statements are then true:


* $ \Pi\_{B,u',v'}: A\_{\theta} \to B $ is the **unique** $ \* $-homomorphism that maps $ [\dot{u}]\_{\mathcal{N}} $ to $ u' $ and $ [\dot{v}]\_{\mathcal{N}} $ to $ v' $.
* $ \Phi\_{A\_{\theta},[\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}}: B \to A\_{\theta} $ is the **unique** $ \* $-homomorphism that maps $ u' $ to $ [\dot{u}]\_{\mathcal{N}} $ and $ v' $ to $ [\dot{v}]\_{\mathcal{N}} $.
* $ \Phi\_{A\_{\theta},[\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}} \circ \Pi\_{B,u',v'}: A\_{\theta} \to A\_{\theta} $ equals $ \text{id}\_{A\_{\theta}} $ on a dense subset of $ A\_{\theta} $.
* $ \Pi\_{B,u',v'} \circ \Phi\_{A\_{\theta},[\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}}: B \to B $ equals $ \text{id}\_{B} $ on a dense subset of $ B $.


Therefore, $ \Phi\_{A\_{\theta},[\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}} \circ \Pi\_{B,u',v'} = \text{id}\_{A\_{\theta}} $ and $ \Pi\_{B,u',v'} \circ \Phi\_{A\_{\theta},[\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}} = \text{id}\_{B} $, and so $ A\_{\theta} $ is $ \* $-isomorphic to $ B $ via the **unique** $ \* $-isomorphism that sends $ [\dot{u}]\_{\mathcal{N}} $ to $ u' $ and $ [\dot{v}]\_{\mathcal{N}} $ to $ v' $.


This concludes the proof that $ (A\_{\theta},[\dot{u}]\_{\mathcal{N}},[\dot{v}]\_{\mathcal{N}}) $ is indeed a universal $ C^{\*} $-representation of $ \mathcal{A}\_{\theta} $. $ \quad \blacksquare $




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The proof that $ \Pi\_{\mathscr{B}({L^{2}}(\mathbb{T})),U,V}: A\_{\theta} \to {C^{\*}}(U,V) $ is a $ \* $-isomorphism follows from the non-trivial fact that $ A\_{\theta} $ is a simple $ C^{\*} $-algebra (i.e., it has no non-trivial closed proper two-sided ideals). The main idea behind the proof is to use the so-called **trace function** on $ A\_{\theta} $.


This trace function does wonders for us. Firstly, it shows that $ A\_{\theta} $ contains a non-trivial projection element. Secondly, it shows that $ \mathcal{A}\_{\theta} $ is faithfully represented as a $ \mathbb{C} $-algebra in $ A\_{\theta} $, i.e., $ \mathcal{N} = \{ 0\_{\mathcal{A}\_{\theta}} \} $. Observe that in defining the $ C^{\*} $-semi-norm $ \| \cdot \|\_{0} $, there was no guarantee that each non-zero element of $ \mathcal{A}\_{\theta} $ would not be sent by $ \| \cdot \|\_{0} $ to $ 0 $. Playing around with the trace function shows that this is indeed the case.


