On the method of constructing irreducible finite index subfactors of popa

Let US(Q) be the universal Jones algebra associated to a finite von Neumann algebra Q and R^s ⊂ R be the Jones subfactors, s ∈ {4cos² πn/|n ≥ 3} ∪ [4, ∞). We consider for any von Neumann subalgebra Q₀ ⊂ Q the algebra U^S(Q, Q₀) defined as the quotient of U^S(Q) through its ideal generated by [Q₀, R] and we construct a Markov trace on U^S(Q, Q₀). If E(Q) ∩ E(Q₀) = C and Q contains n ≥ s + I unitaries u¹ = 1, u₂, …, u_n, with E_Q0(u_i*U_j) = δ_ij1, 1 ≤ i, j ≤ n, then we get a family of irreducible inclusions of type IIi factors N^s ⊂ M^s, with [M^s: N^s] = s and minimal higher relative commutant. Although these subfactors are nonhyperfinite, they have the Haagerup approximation property whether Q₀ ⊂ Q is a Haagerup inclusion and if either Q₀ is finite dimensional or Q₀ ⊂ E(Q).