Subfactor

In the theory of von Neumann algebras, a subfactor of a factor M is a subalgebra that is a factor and contains 1. The theory of subfactors led to the discovery of the Jones polynomial in knot theory.

Index of a subfactor

Usually M is taken to be a factor of type II₁, so that it has a finite trace. In this case every Hilbert space module H has a dimension dim_M(H) which is a non-negative real number or +∞. The index [M:N] of a subfactor N is defined to be dim_N(L²(M)). Here L²(M) is the representation of N obtained from the GNS construction of the trace of M.

The Jones index theorem

This states that if N is a subfactor of M (both of type II₁) then the index [M:N] is either of the form 4 cos(π/n)² for n = 3,4,5,..., or is at least 4. All these values occur.

The first few values of 4 cos(π/n)² are 1, 2, (3+√5)/2=2.618..., 3, 3.247..., ...

The basic construction

Suppose that N is a subfactor of M, and that both are finite von Neumann algebras. The GNS construction produces a Hilbert space L²(M) acted on by M with a cyclic vector Ω. Let e_N be the projection onto the subspace NΩ. Then M and e_N generate a new von Neumann algebra <M, e_N> acting on L²(M), containing M as a subfactor. The passage from the inclusion of N in M to the inclusion of M in <M, e_N> is called the basic construction.

If N and M are both factors of type II₁ and N has finite index in M then <M, e_N> is also of type II₁. Moreover the inclusions have the same index: [M:N] = [<M, e_N> :M], and tr_{<M, eN>}(e_N) = 1/[M:N].

The tower

Suppose that M_-1⊆M_{0 is an inclusion of type II1 factors of finite index. By iterating the basic construction we get a tower of inclusions}

M_-1⊆M_{0⊆M1⊆M2...}

where each M_n+1=<M_n, e_n+1> is generated by the previous algebra and a projection. The union of all these algebras has a tracial state whose restriction to each M_n is the tracial state, and so the closure of the union is another type II₁ von Neumann algebra M_∞.

The algebra M_∞ contains a sequence of projections e₁,e₂, e₃,... and these have the following properties (where we write τ for [M_0:M-1]):

• e_n²=e_n =e_n^*

• tr(e_n) = τ, and more generally tr(xe_n) = τtr(x) for x in M_n-1

• e_me_n=e_ne_m if |m-n|>1.

• e_n+1e_ne_n+1= τe_n+1

• e_ne_n+1e_n= τe_n

Knot polynomials

The algebra generated by the elements e_n 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 given invariants for knots.

Further reading

• V. Jones, V. S. Sunder, Introduction to subfactors, ISBN 0521584205
• Theory of Operator Algebras III by M. Takesaki ISBN 3540429131