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⊆M₀ is an inclusion of type II₁ factors of finite index. By iterating the basic construction we get a tower of inclusions

M_-1⊆M₀⊆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 tr 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,..., which satisfy the Temperley-Lieb relations at parameter λ=1/[M:N]. Moreover, the algebra generated by the e_n is a C*-algebra in which the e_n are self-adjoint, and such that tr(xe_n)=λtr(x) when x is in the algebra generated by e₁ up to e_n-1. Whenever these extra conditions are satisfied, the algebra is called a Temperly-Lieb-Jones algebra at parameter λ. It can be shown to be unique up to *-isomorphism. It exists only when λ takes on those special values 4 cos(π/n)² for n = 3,4,5,..., or the values larger than 4.

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.

References

• Jones, V.F.R. (1983), "Index for subfactors", Invent. Math., 72: 1–25
• Wenzl, H.G. (1988), "Hecke algebras of type A_n and subfactors", Invent. Math., 92: 349–383
• V. Jones, V. S. Sunder, Introduction to subfactors, ISBN 0-521-58420-5
• Theory of Operator Algebras III by M. Takesaki ISBN 3-540-42913-1
• A. J. Wassermann, Operators on Hilbert space