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On the method of constructing irreducible finite index subfactors of Popa. (English) Zbl 0795.46044

Summary: Let \(U^ s(Q)\) be the universal Jones algebra associated to a finite von Neumann algebra \(Q\) and \(R^ s\subset R\) be the Jones subfactors, \(s\in \{4\cos^ 2 {\pi\over n}\mid n\geq 3\}\cup [4,\infty)\). We consider for any von Neumann subalgebra \(Q_ 0\subset Q\) the algebra \(U^ s(Q,Q_ 0)\) defined as the quotient of \(U^ s(Q)\) through its ideal generated by \([Q_ 0,R]\) and we construct a Markov trace on \(U^ s(Q,Q_ 0)\). If \({\mathcal Z}(Q)\cap {\mathcal Z}(Q_ 0)= \mathbb{C}\) and \(Q\) contains \(n\geq s+1\) unitaries \(u_ 1=1\), \(u_ 2,\dots,u_ n\), with \(E_{Q_ 0} (u_ i^* u_ j)= \delta_{ij}1\), \(1\leq i,j\leq n\), then we get a family of irreducible inclusions of type \(II_ 1\) factors \(N^ s\subset M^ s\), with \([M^ s:N^ s]=s\) and minimal higher relative commutant. Although these subfactors are nonhyperfinite, they have the Haagerup approximation property whether \(Q_ 0\subset Q\) is a Haagerup inclusion and if either \(Q_ 0\) is finite dimensional or \(Q_ 0\subset {\mathcal Z}(Q)\).

MSC:

46L37 Subfactors and their classification
46L35 Classifications of \(C^*\)-algebras
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