In the theory of von Neumann algebras, a subfactor of a factor is a subalgebra that is a factor and contains . The theory of subfactors led to the discovery of the Jones polynomial in knot theory.

Index of a subfactor

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Usually   is taken to be a factor of type  , so that it has a finite trace. In this case every Hilbert space module   has a dimension   which is a non-negative real number or  . The index   of a subfactor   is defined to be  . Here   is the representation of   obtained from the GNS construction of the trace of  .

Jones index theorem

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This states that if   is a subfactor of   (both of type  ) then the index   is either of the form   for  , or is at least  . All these values occur.

The first few values of   are  

Basic construction

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Suppose that   is a subfactor of  , and that both are finite von Neumann algebras. The GNS construction produces a Hilbert space   acted on by   with a cyclic vector  . Let   be the projection onto the subspace  . Then   and   generate a new von Neumann algebra   acting on  , containing   as a subfactor. The passage from the inclusion of   in   to the inclusion of   in   is called the basic construction.

If   and   are both factors of type   and   has finite index in   then   is also of type  . Moreover the inclusions have the same index:   and  .

Jones tower

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Suppose that   is an inclusion of type   factors of finite index. By iterating the basic construction we get a tower of inclusions

 

where   and  , and each   is generated by the previous algebra and a projection. The union of all these algebras has a tracial state   whose restriction to each   is the tracial state, and so the closure of the union is another type   von Neumann algebra  .

The algebra   contains a sequence of projections   which satisfy the Temperley–Lieb relations at parameter  . Moreover, the algebra generated by the   is a  -algebra in which the   are self-adjoint, and such that   when   is in the algebra generated by   up to  . Whenever these extra conditions are satisfied, the algebra is called a Temperly–Lieb–Jones algebra at parameter  . It can be shown to be unique up to  -isomorphism. It exists only when   takes on those special values   for  , or the values larger than  .

Standard invariant

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Suppose that   is an inclusion of type   factors of finite index. Let the higher relative commutants be   and  .

The standard invariant of the subfactor   is the following grid:

 
 
 

which is a complete invariant in the amenable case.[1] A diagrammatic axiomatization of the standard invariant is given by the notion of planar algebra.

Principal graphs

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A subfactor of finite index   is said to be irreducible if either of the following equivalent conditions is satisfied:

  •   is irreducible as an   bimodule;
  • the relative commutant   is  .

In this case   defines a   bimodule   as well as its conjugate   bimodule  . The relative tensor product, described in Jones (1983) and often called Connes fusion after a prior definition for general von Neumann algebras of Alain Connes, can be used to define new bimodules over  ,  ,   and   by decomposing the following tensor products into irreducible components:

 

The irreducible   and   bimodules arising in this way form the vertices of the principal graph, a bipartite graph. The directed edges of these graphs describe the way an irreducible bimodule decomposes when tensored with   and   on the right. The dual principal graph is defined in a similar way using   and   bimodules.

Since any bimodule corresponds to the commuting actions of two factors, each factor is contained in the commutant of the other and therefore defines a subfactor. When the bimodule is irreducible, its dimension is defined to be the square root of the index of this subfactor. The dimension is extended additively to direct sums of irreducible bimodules. It is multiplicative with respect to Connes fusion.

The subfactor is said to have finite depth if the principal graph and its dual are finite, i.e. if only finitely many irreducible bimodules occur in these decompositions. In this case if   and   are hyperfinite, Sorin Popa showed that the inclusion   is isomorphic to the model

 

where the   factors are obtained from the GNS construction with respect to the canonical trace.

Knot polynomials

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The algebra generated by the elements   with the relations above is called the Temperley–Lieb algebra. This is a quotient of the group algebra of the braid group, so representations of the Temperley–Lieb algebra give representations of the braid group, which in turn often give invariants for knots.

References

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  1. ^ Popa, Sorin (1994), "Classification of amenable subfactors of type II", Acta Mathematica, 172 (2): 163–255, doi:10.1007/BF02392646, MR 1278111