Universal construction of subfactors (Q2769415)

To each finite index inclusion \(N\subset M\) of factors of type \(II_1\), one can associate its Jones tower \(N\subset M_0=M\subset M_1 \subset M_2 \subset \dots\) obtained by iterating the Jones basic construction. One gets the system \((M_i^\prime \cap M_j)_{0\leq i\leq j}\) of inclusions (\(M_i^\prime \cap M_j \subset M_i^\prime \cap M_{j+1}\) and \(M_i^\prime \cap M_j \subset M_{i-1}^\prime \cap M_j\)) of finite dimensional \(C^*\)-algebras, consisting in the higher relative commutants of the factors in the Jones tower. This system \({\mathcal G}={\mathcal G}_{N,M}\) is called the standard invariant of the inclusion \(N\subset M\) and is endowed with a rich algebraic-combinatorial structure. It plays a fundamental role in the classification of subfactors. NEWLINENEWLINENEWLINEIn a previous paper [Invent. Math. 120, No.~3, 427-445 (1995; Zbl 0831.46069)], the author succeeded in giving an abstract characterization for abstract systems \((A_{i,j})_{0\leq i\leq j}\) of finite dimensional \(C^*\)-algebras, called standard \(\lambda\)-lattices, to arise as lattices of higher relative commutants of irreducible (or more generally extremal) inclusions of given index \(\lambda^{-1}=[M:N] \in \{ 4\cos^2 \frac{\pi}{n} ; n=4,5,\dots\} \cup [4,\infty)\) of factors of type \(II_1\). Extremal subfactors with prescribed standard invariant were produced using an ingenious construction involving reduced amalgamated free product von Neumann algebras. This axiomatization of the lattice of higher relative commutants proved to be beneficial for the theory of subfactors, being used in several instances to check that various systems of algebras arise from subfactors, and to prove new combinatorial results on subfactors and on Jones planar algebras. In the paper under review, the axioms that define a standard \(\lambda\)-lattice are slightly modified. This is done in order to accomodate the non-extremal situation as well. Based on this new and more flexible set of axioms and on his powerful methods developed in earlier work, the author constructs approximately inner, centrally free subfactors, and also subfactors that contain an arbitrary sequence of subfactors as commuting square embeddings, with prescribed standard invariant.