Subfactor

From Wikipedia, the free encyclopedia

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.

Contents

[edit] 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 dimM(H) which is a non-negative real number or +∞. The index [M:N] of a subfactor N is defined to be dimN(L2(M)). Here L2(M) is the representation of N obtained from the GNS construction of the trace of M.

[edit] 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..., ...

[edit] 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 eN be the projection onto the subspace NΩ. Then M and eN generate a new von Neumann algebra <M, eN> 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, eN> is called the basic construction.

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

[edit] 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 Mn+1=<Mn, en+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 Mn 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 en is a C*-algebra in which the en are self-adjoint, and such that tr(xen)=λtr(x) when x is in the algebra generated by e1 up to en-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.

[edit] Knot polynomials

The algebra generated by the elements en 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.

[edit] References