– G 1 ' H k+k0−1 ⊂ H k−1 ⊗ H k0 ⊂ L, – G 2 ' H k+k0−1 ⊂ H 1 ⊗ H k+k0 ⊂ L.

Irreducible corepresentations of A _o (Q) : u _k ∈ B (H _k ) ⊗ A _o (Q), k ∈ N

Quantum dimension of H _k : m _k ∈ R ^∗ ₊

Lemma We consider the following irreducible subspaces of L = H ₁ ⊗ H _k ⊗ H _k

:

– G ₁ ' H _k+k

−1 ⊂ H _k−1 ⊗ H _k

⊂ L, – G ₂ ' H _k+k

−1 ⊂ H ₁ ⊗ H _k+k

⊂ L.

The norm of the orthogonal projection from G ₁ onto G ₂ equals

s

1 − m _k

−1

m _k+k

−1 m _k .

Lemma Let a ∈ S be a coefficient of u ₁ . There exists C _a > 0 such that ∀ k, k ⁰

||[a ⊗ 1, p

](p _k ⊗ p _k

)|| ≤ C _a

s m _k

−1

m _k+k

−1 m _k . NB : m _k+1 − m ₁ m _k + m _k−1 = 0 and

m ₁ = Tr Q ^∗ Q = Tr (Q ^∗ Q) ⁻¹ .