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 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.
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, 51/2, 3, ...
The basic construction
The tower
Knot polynomials
Further reading
- V. Jones, V. S. Sunder, Introduction to subfactors, ISBN 0521584205
- Theory of Operator Algebras III by M. Takesaki ISBN 3540429131