subfactor

A subfactor is an inclusion of von Neumann factors N ⊆ M, where both N and M have trivial centers. In most discussions, the factors are of type II1 and share a common separable structure. The inclusion is studied through the relative size of M over N, captured by the index [M:N], and through the associated tower of algebras obtained by the basic construction.

The Jones index theory associates a numerical index to finite inclusions. If there exists a faithful conditional expectation E: M → N of finite index, then [M:N] is a finite real number with special properties: the index can take values in the set {4 cos^2(π/n) | n = 3,4,5,...} together with all numbers in [4, ∞). In particular, there are only discrete possibilities below 4 and a continuous range above 4. The basic construction yields a Jones tower N ⊆ M ⊆ M1 ⊆ M2 ⊆ ... and a family of relative commutants N′ ∩ Mk that form the standard invariant of the subfactor.

A subfactor is said to have finite depth if its standard invariant is finite, which leads to a finite set of principal graphs encoding the inclusion data. Finite-depth subfactors often correspond to graphs of ADE type and are linked to rich classification results. The concept gives rise to planar algebras and has deep connections to knot theory, notably the Jones polynomial, as well as to quantum groups and conformal field theory.

Examples include inclusions arising from actions of finite groups on II1 factors and their fixed-point subfactors, or crossed products by finite groups. Subfactor theory provides a framework to study symmetry, invariants, and category-theoretic structures in operator algebras.