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Proof. (1) follows immediately from the definitions. By (1), we just have to show (2) in the case that ρ1 , σ1 , ρ2 , σ2 are irreducible sectors. It is obvious that if ρ1 σ1 , ρ2 σ2 as sectors, then ρ1 ⊗ ρ2 σ1 ⊗ σ2 as sectors of N (I ). Now suppose ρ1 ⊗ ρ2 σ1 ⊗ σ2 as sectors of N(I ). This means there exists a unitary u ∈ N (I ) such that: uρ1 (p) ⊗ ρ2 (p ) = σ1 (p) ⊗ σ2 (p )u, for any p ∈ A(I ), p ∈ π0 (LI H ) . By the statement on p. 123 of [Stra], there exists normal conditional expectation E : N (I ) → A(I ) ⊗ 1 such that E(u) = 0.
In fact suppose (i, α) is an irreducible sector. Let p ∈ (∨J ∈I πi,α (A(J ))) . Then p ∈ πi,α (A(I )) = A(I ). It follows that p ∈ Hom((i, α), (i, α)) = C, since (i, α) is irreducible. On the other hand if πi,α is irreducible, and p ∈ Hom((i, α), (i, α)). Then p ∈ A(I ) and so p ∈ (πi,α (A(I )) ∨ πi,α (A(I ))) . But πi,α (A(I )) ∨ πi,α (A(I )) = ∨J ∈I πi,α (A(J )) by the strong additivity of the coset conformal precosheaf, so p ∈ (∨J ∈I πi,α (A(J ))) = C, since πi,α is irreducible. Similarly one can show that (i, α) (j, β) if and only if πj,β appears as a direct summand of πi,α , and (i, α) is equal to (j, β) as sectors if and only if πi,α is unitarily equivalent to πj,β .
7 of [KW]3 . But since d(0,α) = dα , we have: dρ2 = dα dα . α∈exp0 It follows that all the ≤’s above are actually =, in particular di,α = di dα , and it follows from (2) of Prop. 3 that a(i,α)⊗1 = σi a1⊗α¯ . 6): dσ (0,0) = dσ (0) dσ (0) = 1. On the other hand σ (0, 0), (0, 0) ≥ 1, it follows by comparing statistical dimensions that σ (0, 0), (0, 0) = 1. So we can improve (**) to a1⊗α¯ σi , a1⊗α σi = δσ (i),i δσ (α),α . (∗) σ ∈ZN Ad (2): By assumption expression (*) holds for a unique σ since the action is faithful.