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Classification of amenable subfactors of type II. (English) Zbl 0853.46059
A central problem arising in the theory of subfactors is the classification of subfactors $$N\subset M$$ of finite index of the hyperfinite (or approximately finite dimensional) factors $$M$$. The physically relevant invariant for such a subfactor is the lattice of its higher relative commutants $$\{M'_i\cap M_j\}_{i,j}$$ in the Jones tower $$N\subset M\subset M_1\subset\dots$$; this is called the standard invariant of $$N\subset M$$, denoted $$\mathcal G_{N,M}$$. The inclusions between these higher relative commutants are described by Jones’ principal graph $$\Gamma_{N,M}$$. An abstract characterization of the invariant $$\mathcal G_{N,M}$$ is given in [A. Ocneanu, Quantized groups, string algebras, and Galois theory for von Neumann algebras, London Math. Soc. Lecture Notes 136, 119-172 (1988; Zbl 0696.46048] as the paragroup of all irreducible correspondences (bimodules) generated under Connes’ composition (or fusion) rule by $$N\subset M$$, and $$\Gamma_{N,M}$$ as its “fusion rule matrix” (a Cayley-type graph).
The main result of the paper under review is that for a large class of subfactors, called strongly amenable, the standard invariant $$\mathcal G_{N,M}$$ is the complete invariant. It should be noted that all the examples of subfactors coming out this far for quantum field theories and polynomial invariants for knots are strongly amenable. Also, the classical problems of classifying actions of amenable discrete groups and compact Lie groups on hyperfinite factors were given equivalent formulations in terms of classification of certain subfactors that are strongly amenable, in an earlier work by A. Wasserman and the author [Actions of compact Lie groups on von Neumann algebras, C. R. Acad. Sci. Paris Sér. I Math. 315, No. 4, 421-426 (1992; Zbl 0794.46050)]. The result is in some sense the best that can be obtained, as it is shown that strongly amenable subfactors give the largest class of subfactors which can be reconstructed in a direct way from their standard invariants.
The definition of strong amenability goes as follows. First a representation of an inclusion of type II$$_1$$ subfactors $$N\subset M$$ (of finite index) is defined as an embedding of $$N\subset M$$ into an inclusion of von Neumann algebras $$\mathcal N\subset^{\mathcal E}\mathcal M$$, with $$N\subset\mathcal N$$, $$M\subset\mathcal M$$ and with $$\mathcal E:\mathcal M\to\mathcal N$$ a conditional expectation of $$\mathcal M$$ onto $$\mathcal N$$ that restricted to $$\mathcal M$$ agrees with the trace preserving expectation of $$M$$ onto $$N$$. The inclusion $$N\subset M$$ is then called amenable if whenever represented smoothly in some $$\mathcal N\subset^{\mathcal E}\mathcal M$$ there exists an $$M$$-hypertrace $$\phi$$ on $$\mathcal M$$ satisfying $$\phi=\phi\circ\mathcal E$$; this is equivalent to the existence of a norm one projection of $$\mathcal N\subset\mathcal M$$ onto $$N\subset M$$. Finally $$N\subset M$$ is strongly amenable if in addition the graph $$\Gamma_{N,M}$$ is ergodic. Equivalently, $$N\subset M$$ is strongly amenable if and only if it is isomorphic to the canonical model $$N^{\text{st}} \subset M^{\text{st}}$$ coming from the standard invariant $$\mathcal G_{N,M}$$, i.e. from the higher relative commutants. For inclusions of type II$$_\infty$$ factors $$N^\infty\subset M^\infty$$ of finite index, (strong) amenability is defined as the (strong) amenability of the type II$$_1$$ inclusion $$pN^\infty p\subset p M^\infty p$$ obtained by reducing with finite projections in $$N^\infty$$.
Reviewer: M.Engliš (Praha)

##### MSC:
 46L37 Subfactors and their classification
##### Keywords:
operator algebras; factors; Jones tower; amenable subfactors
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