In the theory of von Neumann algebras, a subfactor of a factor
is a subalgebra that is a factor and contains
The theory of subfactors led to the discovery of the Jones polynomial in knot theory.
In this case every Hilbert space module
obtained from the GNS construction of the trace of
, and that both are finite von Neumann algebras.
The GNS construction produces a Hilbert space
generate a new von Neumann algebra
is called the basic construction.
factors of finite index.
By iterating the basic construction we get a tower of inclusions where
is generated by the previous algebra and a projection.
The union of all these algebras has a tracial state
is the tracial state, and so the closure of the union is another type
factors of finite index.
is the following grid: which is a complete invariant in the amenable case.
[1] A diagrammatic axiomatization of the standard invariant is given by the notion of planar algebra.
is said to be irreducible if either of the following equivalent conditions is satisfied: In this case
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
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
The dual principal graph is defined in a similar way using
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.
are hyperfinite, Sorin Popa showed that the inclusion
factors are obtained from the GNS construction with respect to the canonical trace.
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.