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.

$C^*$-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 $\bigoplus_{i}L^2(S^1)_i$? $\endgroup$