In the theory of von Neumann algebras, a subfactor of a factor formula_1 is a subalgebra that is a factor and contains formula_2. The theory of subfactors led to the discovery of the 
Jones polynomial in knot theory.
Index of a subfactor.
Usually formula_1 is taken to be a factor of type formula_4, so that it has a finite trace.
In this case every Hilbert space module formula_5 has a dimension formula_6 which is a non-negative real number or formula_7. 
The index formula_8 of a subfactor formula_9 is defined to be formula_10. Here formula_11 is the representation 
of formula_9 obtained from the GNS construction of the trace of formula_1.
Jones index theorem.
This states that if formula_9 is a subfactor of formula_1 (both of type formula_4) then the index formula_17 is either of the form formula_18 for formula_19, or is at least formula_20. All these values occur.
The first few values of formula_18 are formula_22
Basic construction.
Suppose that formula_9 is a subfactor of formula_1, and that both are finite von Neumann algebras. 
The GNS construction produces a Hilbert space formula_25 acted on by formula_1
with a cyclic vector formula_27. Let formula_28 be the projection onto the subspace formula_29. Then formula_1 and formula_28 generate a new von Neumann algebra formula_32 acting on formula_11, containing formula_34 as a subfactor. The passage from the inclusion of formula_35 in formula_34 to the inclusion of formula_34 in formula_32 is called the basic construction.
If formula_35 and formula_34 are both factors of type formula_4 and formula_35 has finite index in formula_34 then formula_32 is also of type formula_4.
Moreover the inclusions have the same index: formula_46 and formula_47.
Jones tower.
Suppose that formula_48 is an inclusion of type formula_4 factors of finite index. By iterating the basic construction we get a tower of inclusions
 formula_50
where formula_51 and formula_52, and each formula_53 is generated by the previous algebra and a projection. The union of all these algebras has a tracial state formula_54 whose restriction to each formula_55 is the tracial state, and so the closure of the union is another type formula_4 von Neumann algebra formula_57.
The algebra formula_57 contains a sequence of projections formula_59 which satisfy the Temperley–Lieb relations at parameter formula_60. Moreover, the algebra generated by the formula_61 is a formula_62-algebra in which the formula_61 are self-adjoint, and such that formula_64 when formula_65 is in the algebra generated by formula_66 up to formula_67. Whenever these extra conditions are satisfied, the algebra is called a Temperly–Lieb–Jones algebra at parameter formula_68. It can be shown to be unique up to formula_69-isomorphism. It exists only when formula_68 takes on those special values formula_71 for formula_19, or the values larger than formula_20.
Standard invariant.
Suppose that formula_48 is an inclusion of type formula_4 factors of finite index. Let the higher relative commutants be formula_76 and formula_77.
The standard invariant of the subfactor formula_48 is the following grid:
 formula_79 
 formula_80 
 formula_81 
which is a complete invariant in the amenable case. A diagrammatic axiomatization of the standard invariant is given by the notion of planar algebra.
Principal graphs.
A subfactor of finite index formula_82 is said to be irreducible if either of the following equivalent conditions is satisfied:
In this case formula_83 defines a formula_84 bimodule formula_89 as well as its conjugate formula_90 bimodule formula_91. The relative tensor product, described in 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 formula_84, formula_90, formula_94 and formula_95 by decomposing the following tensor products into irreducible components:
formula_96
The irreducible formula_94 and formula_90 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 formula_99 and formula_91 on the right. 
The dual principal graph is defined in a similar way using formula_95 and formula_84 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 formula_34 and formula_35 are hyperfinite, Sorin Popa showed that the inclusion formula_105 is isomorphic to the model
formula_106
where the formula_4 factors are obtained from the GNS construction with respect to the canonical trace.
Knot polynomials.
The algebra generated by the elements formula_61 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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