Subfactor Explained
is a subalgebra that is a factor and contains
. The theory of subfactors led to the discovery of the
Jones polynomial in
knot theory.
Index of a subfactor
Usually
is taken to be a factor of type
, so that it has a finite trace.In this case every Hilbert space module
has a dimension
which is a non-negative real number or
. The
index
of a subfactor
is defined to be
. Here
is the representation of
obtained from the
GNS construction of the trace of
.
Jones index theorem
This states that if
is a subfactor of
(both of type
) then the index
is either of the form
for
, or is at least
. All these values occur.
The first few values of
are
1,2,(3+\sqrt{5})/2=2.618...,3,3.247...,...
Basic construction
Suppose that
is a subfactor of
, and that both are finite von Neumann algebras. The GNS construction produces a Hilbert space
acted on by
with a cyclic vector
. Let
be the projection onto the subspace
. Then
and
generate a new von Neumann algebra
acting on
, containing
as a subfactor. The passage from the inclusion of
in
to the inclusion of
in
is called the
basic construction.
If
and
are both factors of type
and
has finite index in
then
is also of type
.Moreover the inclusions have the same index:
[M:N]=[\langleM,eN\rangle:M],
and
.
Jones tower
Suppose that
is an inclusion of type
factors of finite index. By iterating the basic construction we get a tower of inclusions
M-1\subsetM0\subsetM1\subsetM2\subset …
where
and
, and each
Mn+1=\langleMn,en+1\rangle
is generated by the previous algebra and a projection. The union of all these algebras has a tracial state
whose restriction to each
is the tracial state, and so the closure of the union is another type
von Neumann algebra
.
The algebra
contains a sequence of projections
which satisfy the Temperley - Lieb relations at parameter
. Moreover, the algebra generated by the
is a
-algebra in which the
are self-adjoint, and such that
when
is in the algebra generated by
up to
. Whenever these extra conditions are satisfied, the algebra is called a Temperly - Lieb - Jones algebra at parameter
. It can be shown to be unique up to
-isomorphism. It exists only when
takes on those special values
for
, or the values larger than
.
Standard invariant
Suppose that
is an inclusion of type
factors of finite index. Let the higher relative commutants be
and
.
The standard invariant of the subfactor
is the following grid:
C=l{P}0,+\subsetl{P}1,+\subsetl{P}2,+\subset … \subsetl{P}n,+\subset …
C =l{P}0,-\subsetl{P}1,-\subset … \subsetl{P}n-1,-\subset …
which is a complete invariant in the amenable case. A diagrammatic axiomatization of the standard invariant is given by the notion of
planar algebra.
Principal graphs
A subfactor of finite index
is said to be
irreducible if either of the following equivalent conditions is satisfied:
is irreducible as an
bimodule;
is
.
In this case
defines a
bimodule
as well as its conjugate
bimodule
. The relative tensor product, described in 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
,
,
and
by decomposing the following tensor products into irreducible components:
X\boxtimesX\star\boxtimes … \boxtimesX,X\star\boxtimesX\boxtimes … \boxtimesX\star,X\star\boxtimesX\boxtimes … \boxtimesX,X\boxtimesX\star\boxtimes … \boxtimesX\star.
The irreducible
and
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
and
on the right. The
dual principal graph is defined in a similar way using
and
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
and
are hyperfinite,
Sorin Popa showed that the inclusion
is isomorphic to the model
(C ⊗ EndX\star\boxtimesX\boxtimesX\star\boxtimes … )\prime\prime\subset(EndX\boxtimesX\star\boxtimesX\boxtimesX\star\boxtimes … )\prime\prime,
where the
factors are obtained from the GNS construction with respect to the canonical trace.
Knot polynomials
The algebra generated by the elements
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
- Book: Jones . Vaughan F.R.. Vaughan Jones. Sunder. Viakalathur Shankar. Introduction to subfactors. London Mathematical Society Lecture Note Series. 234. 1997. Cambridge University Press. Cambridge. 10.1017/CBO9780511566219. 0-521-58420-5. 1473221.
- Theory of Operator Algebras III by M. Takesaki
- Web site: Antony. Wassermann. Antony Wassermann. Operators on Hilbert space.