Greg Kuperberg gave a great reply without explaining what a von Neumann algebra is, but von Neumann algebras are really not bad. In particular, the type $II_1$ hyperfinite factor used by Vaughan Jones work is a very nice algebra — apart from some fine print, it’s the algebra generated by creation and annihilation oeprators in any free field theory of fermions.

So: what’s a von Neumann algebra? Before I get technical and you all
leave, I should just say that von Neumann designed these algebras to be
good "algebras of observables" in quantum theory. The
simplest example consists of all n x n complex matrices: these become an
algebra if you add and multiply them the usual way. So, the subject of
von Neumann algebras is really just a grand generalization of the theory
of matrix multiplication.

But enough beating around the bush! For starters, a von Neumann algebra
is a *-algebra of bounded operators on some Hilbert space of countable
dimension - that is, a bunch of bounded operators closed under addition,
multiplication, scalar multiplication, and taking adjoints: that’s the *
business. However, to be a von Neumann algebra, our *-algebra needs one
extra property! This extra property is cleverly chosen so that we can
apply functions to observables and get new observables, which is
something we do all the time in physics.

More precisely, given any self-adjoint operator $A$ in our von Neumann
algebra and any measurable function $f: \mathbb{R} \to \mathbb{R}$ we want there to be a
self-adjoint operator $f(A)$ that again lies in our von Neumann algebra.
To make sure this works, we need our von Neumann algebra to be "closed"
in a certain sense. The nice thing is that we can state this closure
property either algebraically or topologically.

In the algebraic approach, we define the "commutant" of a bunch of
operators to be the set of operators that commute with all of them.
We then say a von Neumann algebra is a *-algebra of operators that’s
the commutant of its commutant.

The commutant of the commutant is “everyone who commutes with everyone who commutes with you”. It’s a bit like “the enemy of my enemy is my friend” — the double commutant of any bunch of operators includes all reasonable functions of those operators: their “friends”.

In the topological approach, we say a bunch of operators $T_i$
converges "weakly" to an operator $T$ if their expectation
values converge to that of $T$ in every state, that is,

$\langle \psi, T_i \psi \rangle \to \langle \psi, T \psi \rangle$

for all unit vectors $\psi$ in the Hilbert space. We then say a von
Neumann algebra is an *-algebra of operators that is closed in the
weak topology.

It’s a nontrivial theorem that these two definitions agree!

While classifying all *-algebras of operators is an utterly hopeless
task, classifying von Neumann algebras is almost within reach - close
enough to be tantalizing, anyway. Every von Neumann algebra can be
built from so-called "simple" ones as a direct sum, or more generally a
"direct integral", which is a kind of continuous version of a direct
sum. As usual in algebra, the "simple" von Neumann algebras are defined
to be those without any nontrivial ideals. This turns out to be
equivalent to saying that only scalar multiples of the identity commute
with everything in the von Neumann algebra.

People call simple von Neumann algebras "factors" for short. Anyway,
the point is that we just need to classify the factors: the process
of sticking these together to get the other von Neumann algebras is
not tricky.

The first step in classifying factors was done by von Neumann and
Murray, who divided them into types I, II, and III. This classification
involves the concept of a "trace", which is a generalization
of the usual trace of a matrix.

Here’s the definition of a trace on a von Neumann algebra. First, we
say an element of a von Neumann algebra is "nonnegative" if
it’s of the form xx* for some element x. The nonnegative elements form
a "cone": they are closed under addition and under
multiplication by nonnegative scalars. Let P be the cone of nonnegative
elements. Then a "trace" is a function

$tr: P \to [0, +\infty]$

which is linear in the obvious sense and satisfies

$tr(xy) = tr(yx)$

whenever both xy and yx are nonnegative.

Note: we allow the trace to be infinite, since the interesting von
Neumann algebras are infinite-dimensional. This is why we define
the trace only on nonnegative elements; otherwise we get "infinity minus
infinity" problems. The same thing shows up in the measure theory,
where we start by integrating nonnegative functions, possibly getting
the answer $+\infty$, and worry later about other functions.

Indeed, a trace very much like an integral, so we’re really studying a
noncommutative version of the theory of integration. On the other hand,
in the matrix case, the trace of a projection operator is just the
dimension of the space it’s the projection onto. We can define a
"projection" in any von Neumann algebra to be an operator with
p* = p and p^{2} = p. If we study the trace of such a thing,
we’re studying a *generalization of the concept of dimension*.
It turns out this can be infinite, or even nonintegral!

We say a factor is type I if it admits a nonzero trace for
which the trace of a projection lies in the set $\{0,1,2,...,+\infty\}$.
We say it’s type I_{n} if we can normalize the trace
so we get the values {0,1,…,n}. Otherwise, we say it’s type
I_{∞}, and we can normalize the trace to get all the
values $\{0,1,2,...,+\infty\}$.

It turns out that every type I_{n} factor is isomorphic to the
algebra of $n \times n$ matrices. Also, every type I_{infinity} factor
is isomorphic to the algebra of all bounded operators on a Hilbert space
of countably infinite dimension.

In short, type I factors are the algebras of observables that we learn to love in
quantum mechanics. So, the real achievement of von Neumann was to begin
exploring the other factors, which turned out to be important in quantum
field theory.

We say a factor is type II_{1} if it admits a trace
whose values on projections are all the numbers in the unit interval
[0,1]. We say it is type II_{∞} if it admits
a trace whose value on projections is everything in [0,+∞].

Playing with type II factors amounts to letting dimension be a
continuous rather than discrete parameter!

Weird as this seems, it’s easy to construct a type II_{1}
factor. Start with the algebra of 1 × 1 matrices, and stuff it into the
algebra of 2 × 2 matrices as follows:

$x \mapsto \left(
\array{
x & 0 \\
0 & x
} \right)$

This doubles the trace, so define a new trace on the algebra of 2 × 2
matrices which is half the usual one. Now keep doing this, doubling the
dimension each time, using the above formula to define a map from the
2^{n} × 2^{n} matrices into the 2^{n+1} ×
2^{n+1} matrices, and normalizing the trace on each of these
matrix algebras so that all the maps are trace-preserving. Then take
the *union* of all these algebras… and finally, with a little work, complete this and get a von Neumann algebra!

One can show this von Neumann algebra is a factor. It’s pretty
obvious that the trace of a projection can be any fraction in the
interval [0,1] whose denominator is a power of two. But actually,
*any* number from 0 to 1 is the trace of some projection in this
algebra - so we’ve got our paws on a type II_{1} factor.

This isn’t the only II_{1} factor, but it’s the only one that
contains a sequence of finite-dimensional von Neumann algebras whose
union is dense in the weak topology. A von Neumann algebra like that is
called "hyperfinite", so this guy is called "the
hyperfinite II_{1} factor".

It may sound like something out of bad science fiction, but the
hyperfinite II_{1} factor shows up all over the place in physics!

First of all, the algebra of 2^{n} × 2^{n} matrices is a
Clifford algebra, so the hyperfinite II_{1} factor is a kind of
infinite-dimensional Clifford algebra. But the Clifford algebra of
2^{n} × 2^{n} matrices is secretly just another name for
the algebra generated by creation and annihilation operators on the
fermionic Fock space over $\mathbb{C}^{2n}$. Pondering this a bit, you can show
that the hyperfinite II_{1} factor is the smallest von Neumann
algebra containing the creation and annihilation operators on a
fermionic Fock space of countably infinite dimension.

In less technical lingo - I’m afraid I’m starting to assume you know
quantum field theory! - the hyperfinite II_{1} factor is the
right algebra of observables for a free quantum field theory with only
fermions. For bosons, you want the type I_{∞} factor.

There is more than one type II_{∞} factor, but again
there is only one that is hyperfinite. You can get this by tensoring
the type I_{∞} factor and the hyperfinite II_{1} factor. Physically, this means that the hyperfinite
II_{∞} factor is the right algebra of observables for a
free quantum field theory with both bosons and fermions!

The most mysterious factors are those of type III. These can be simply
defined as "none of the above"! Equivalently, they are factors for
which any nonzero trace takes values in the two-element set {0,∞}. In a type III
factor, all projections other than 0 have infinite trace. In other
words, the trace is a useless concept for these guys.

As far as I’m concerned, the easiest way to construct a type III factor
uses physics. Now, I said that free quantum field theories had
different kinds of type I or type II factors as their algebras of
observables. This is true if you consider the algebra of *all*
observables. However, if you consider a free quantum field theory on
(say) Minkowski spacetime, and look only at the observables that you can
cook from the field operators on some bounded open set, you get a
subalgebra of observables which turns out to be a type III factor!

In fact, this isn’t just true for free field theories. According to a
theorem of axiomatic quantum field theory, pretty much all the usual
field theories on Minkowski spacetime have type III factors as their
algebras of "local observables" - observables that can be measured in
a bounded open set.

## A Response from Greg Kuperberg

Greg Kuperberg answered my plea via email and — with his permission — I’m posting his response here.

The main construction of Asaeda and Haagerup actually makes no direct use of von Neumann algebras and you don’t need to know anything about the latter to understand the former. So your plea is really two different questions: (1) what did Asaeda and Haagerup do; and (2) what are von Neumann algebras, in physics-speak?

What Asaeda and Haagerup constructed is a fusion category. This is an abstract algebraic object that satisfies many of the Moore-Seiberg axioms for the conformal blocks of a conformal field theory. Conformality plays no direct role either (as you can see from the Moore-Seiberg paper); the axioms really describe a topological field theory. To be specific, the axioms require:

Note that no braiding is required in these axioms, unlike in Moore and Seiberg. However, there is a remarkable general construction, called the double of a fusion category, that is twice as big in a natural sense and that is automatically braided. Indeed, the double of a finite fusion category is modular and yields a 3D TQFT.

Without the finiteness axiom, fusion categories include the traditional example of spin statistics, as developed by Racah and Wigner and as taught in good quantum mechanics courses. The spin category has a species for each non-negative half-integer $j$ and the $6j$-symbol is the usual one.

Every finite group yields a unitary fusion category (as studied by Dijkgraaf and Witten, among others). So does Chern-Simons theory with any compact gauge group; an equivalent construction uses a quantum group at a root of unity. There are also various clever ways to combine two fusion categories; these combinations resemble the ways that you can combine finite groups.

A species in a fusion category has an intrinsic dimension which can be derived from the Clebsch-Gordan branching rule. The intrinsic dimension is always a real number which is at least 1, but it does not have to be an integer. Some fusion categories have the property that each species can be represented by a finite-dimensional Hilbert space; for example, the traditional Racah-Wigner spin category. If this is possible, then each intrinsic dimension is an integer and is the dimension of the Hilbert space.

The simplest fusion category which cannot be made from any kind of fiddling with finite groups is the Fibonacci category. It appears in Chern-Simons theory four times, using different gauge groups and levels. It has one non-trivial species, whose intrinsic dimension is the golden ratio. It is also universal for quantum computation and much-sought by anyon physicists for that reason. Actually, anyonic statistics in general is described by a braided fusion category; the Chern-Simons construction of the category need not be physical.

Asaeda and Haagerup found two exotic fusion categories that do not come from any previous constructions. Like many such categories of interest to operator algebraists, their fusion category has a distinguished generating species $V$. The intrinsic dimension of the generating species in their two examples is $(5+\sqrt{13})/2$ and $(5+\sqrt{17})/2$, as you report. Also, there are 10 and 8 irreducible species, respectively. The Clebsch-Gordan rule for tensoring with $V$ can be summarized with a graph that connects all of the species, as shown in Figure 1 of their paper.

These intrinsic dimensions do not appear among Chern-Simons TQFTs as they are usually defined; nor do the Clebsch-Gordan graphs in Figure 1. However, any modular, unitary fusion category can be viewed as a generalized Chern-Simons TQFT; and that includes the doubles of the Asaeda-Haagerup categories.

I will save question (2) for later, or skip it. Its connection to the present topic is as follows: As Moore and Seiberg discussed, a conformal field theory has a topological sector given by a (braided or modular) fusion category. But this is just one sector of something else richer and not entirely topological. In the same spirit but earlier, Vaughan Jones discovered by example that if you have an irreducible von Neumann algebra with an irreducible sub-von-Neumann algebra, than that too has a “topological” sector described by a fusion category (which is only sometimes braided). That is how Vaughan discovered his polynomial: he essentially discovered the fusion category of quantum $SU(2)$, although the quantum group and Chern-Simons models came later. It’s also why Asaeda and Haagerup were motivated to look for exotic fusion categories.