Realisation of the noncommutative torus as a universal $ C^{*} $-algebra 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:


*

*as the $ C^{*} $-algebra of a foliation;

*as a crossed-product $ C^{*} $-algebra;

*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?
 A: 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! 
A: 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.

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.


*

*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} $.

*Let $ \mathcal{I}_{\theta} $ denote the (not a priori proper) two-sided ideal
$$
\left\langle
u u^{*} - \mathbf{1},
u^{*} u - \mathbf{1},
v v^{*} - \mathbf{1},
v^{*} v - \mathbf{1},
u v - e^{2 \pi i \theta} v u
\right\rangle.
$$

*Form the quotient $ \mathbb{C} $-algebra $ \mathcal{A}_{\theta} \stackrel{\text{df}}{=} \mathcal{P} / \mathcal{I}_{\theta} $.

*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.

*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.



*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 $.


*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.



*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} $.

*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.


*Notice that $ \| \cdot \|_{0} $ also satisfies the $ C^{*} $-identity.

*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} $.

*Form the quotient $ \mathbb{C} $-algebra $ \mathcal{A}_{\theta} / \mathcal{N} $ to get a pre-$ C^{*} $-algebra, denoting the quotient norm by $ \| \cdot \| $.

*Complete $ \mathcal{A}_{\theta} / \mathcal{N} $ with respect to $ \| \cdot \| $ to obtain the irrational rotation $ C^{*} $-algebra $ A_{\theta} $.

*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.


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 $

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.
