Subfactor

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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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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