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.

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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, (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 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].

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) = λ tr(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.

Principal graphs

A subfactor of finite index N \subseteq M is said to be irreducible is either of the following equivalent conditions are satisfied

In this case L2(M) defines an (N, M) bimodule X as well as its conjugate (M, N) bimodule X*. 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 (N, M), (M, N), (M, M) and (N, N) by decomposing the following tensor products into irreducible components:

X\boxtimes X^* \boxtimes \cdots \boxtimes X,\,\, X^*\boxtimes X \boxtimes \cdots \boxtimes X^*, \,\, X^* \boxtimes X \boxtimes \cdots \boxtimes X,\,\, X\boxtimes X^* \boxtimes \cdots \boxtimes X^*.

The irreducible (M, M) and (M, N) 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 X and X* on the right. The dual principal graph is defined in a similar way using (N, N) and (N, M) 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 M and N are hyperfinite, Sorin Popa showed that the inclusion N \subseteq M is isomorphic to the model

(\mathbf{C}\otimes \mathrm{End}\, X^*\boxtimes X \boxtimes X^*\boxtimes \cdots)^{\prime\prime} \subseteq (\mathrm{End}\, X\boxtimes X^* \boxtimes X \boxtimes X^* \boxtimes\cdots )^{\prime\prime},

where the II1 factors are obtained from the GNS construction with respect to the canonical trace.

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 give invariants for knots.

References