On the method of constructing irreducible finite index subfactors of Popa

Florin Petre Boca

Vol. 161 (1993), No. 2, 201–231

DOI: 10.2140/pjm.1993.161.201

Abstract

Let U^s(Q) be the universal Jones algebra associated to a finite von Neumann algebra Q and R^s ⊂ R be the Jones subfactors, s ∈{4cos²|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 𝒵(Q) ∩𝒵(Q₀) = ℂ and Q contains n ≥ s + 1 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 II₁ 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 ⊂𝒵(Q).

Mathematical Subject Classification 2000

Primary: 46L37

Milestones

Received: 19 December 1991

Revised: 16 July 1992

Published: 1 December 1993