16b. Subfactor theory

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In view of Principle 16.5, and its quantum mechanics ramifications, it looks reasonable to forget about the Hilbert space [math]H[/math], about operators [math]T\in B(H)[/math], about other von Neumann algebras and factors [math]A\subset B(H)[/math] that might appear, about other mathematics and physics too, why not about your friends, spouse and hobbies too, but please keep teaching some calculus, that is first class mathematics, and focus on the [math]{\rm II}_1[/math] factors.

With this idea in mind, we have our objects, the [math]{\rm II}_1[/math] factors, but what about morphisms. And here, a natural idea is that of looking at the inclusions of such factors:

Definition

A subfactor is an inclusion of [math]{\rm II}_1[/math] factors [math]A_0\subset A_1[/math].

So, these will be the objects that we will be interested in, in what follows. With the comment that, while quantum mechanics and von Neumann algebras have been around for a while, since the 1920s, and Definition 16.7 is something very natural emerging from this, it took mathematics and physics a lot of time to realize this, with Definition 16.7 basically dating back to the late 1970s, with the beginning of the work of Jones, on it. Moral of the story, sometimes it takes a lot of skill, to come up with simple things.

Now given a subfactor [math]A_0\subset A_1[/math], a first question is that of defining its index, measuring how big [math]A_1[/math] is, when compared to [math]A_0[/math]. But this can be done as follows:

Theorem

Given a subfactor [math]A_0\subset A_1[/math], the number

[[math]] N=\frac{\dim_{A_0}H}{\dim_{A_1}H} [[/math]]

is independent of the ambient Hilbert space [math]H[/math], and is called index.

Show Proof

This is something quite standard, the idea being as follows:

(1) To start with, given a representation of a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], we can talk about the corresponding coupling constant, as being a number as follows:

[[math]] \dim_AH\in(0,\infty] [[/math]]

To be more precise, we can construct this coupling constant in the following way, with [math]u:H\to L^2(A)\otimes l^2(\mathbb N)[/math] being an isometry satisfying [math]ux=(x\otimes1)u[/math]:

[[math]] \dim_AH=tr(uu^*) [[/math]]

(2) Alternatively, we can use the following formula, after proving first that the number on the right is indeed independent of the choice on a nonzero vector [math]x\in H[/math]:

[[math]] \dim_AH=\frac{tr_A(P_{A'x})}{tr_{A'}(P_{Ax})} [[/math]]

This latter formula was in fact the original definition of the coupling constant, by Murray and von Neumann. However, technically speaking, it is better to use (1).

(3) Now with this in hand, given a subfactor [math]A_0\subset A_1[/math], the fact that the index as defined above is indeed independent of the ambient Hilbert space [math]H[/math] comes from the various basic properties of the coupling constant, established by Murray and von Neumann.

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There are many examples of subfactors coming from groups, and every time we obtain the intuitive index. In general now, following Jones , let us start with:

Proposition

Given a subfactor [math]A_0\subset A_1[/math], there is a unique linear map

[[math]] E:A_1\to A_0 [[/math]]

which is positive, unital, trace-preserving and which is such that, for any [math]a_1,a_2\in A_0[/math]:

[[math]] E(a_1ba_2)=a_1E(b)a_2 [[/math]]

This map is called conditional expectation from [math]A_1[/math] onto [math]A_0[/math].

Show Proof

We make use of the standard representation of the [math]{\rm II}_1[/math] factor [math]A_1[/math], with respect to its unique trace [math]tr:A_1\to\mathbb C[/math], namely:

[[math]] A_1\subset L^2(A_1) [[/math]]

If we denote by [math]\Omega[/math] the standard cyclic and separating vector of [math]L^2(A_1)[/math], we have an identification of vector spaces [math]A_0\Omega=L^2(A_0)[/math]. Consider now the following projection:

[[math]] e:L^2(A_1)\to L^2(A_0) [[/math]]

It follows from definitions that we have an inclusion [math]e(A_1\Omega)\subset A_0\Omega[/math]. Thus the above projection [math]e[/math] induces by restriction a certain linear map, as follows:

[[math]] E:A_1\to A_0 [[/math]]

This linear map [math]E[/math] and the orthogonal projection [math]e[/math] are related by:

[[math]] exe=E(x)e [[/math]]

But this shows that the linear map [math]E[/math] satisfies the various conditions in the statement, namely positivity, unitality, trace preservation and bimodule property. As for the uniqueness assertion, this follows by using the same argument, applied backwards, the idea being that a map [math]E[/math] as in the statement must come from a projection [math]e[/math].

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We will be interested in what follows in the orthogonal projection [math]e:L^2(A_1)\to L^2(A_0)[/math] producing the expectation [math]E:A_1\to A_0[/math], rather than in [math]E[/math] itself:

Definition

Associated to any subfactor [math]A_0\subset A_1[/math] is the orthogonal projection

[[math]] e:L^2(A_1)\to L^2(A_0) [[/math]]

producing the conditional expectation [math]E:A_1\to A_0[/math] via the following formula:

[[math]] exe=E(x)e [[/math]]

This projection is called Jones projection for the subfactor [math]A_0\subset A_1[/math].

Quite remarkably, the subfactor [math]A_0\subset A_1[/math], as well as its commutant, can be recovered from the knowledge of this projection, in the following way:

Proposition

Given a subfactor [math]A_0\subset A_1[/math], with Jones projection [math]e[/math], we have

[[math]] A_0=A_1\cap\{e\}'\quad,\quad A_0'=(A_1'\cap\{e\})'' [[/math]]

as equalities of von Neumann algebras, acting on the space [math]L^2(A_1)[/math].

Show Proof

The above two formulae both follow from [math]exe=E(x)e[/math], via some elementary computations, and for details here, we refer to Jones' paper .

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We are now ready to formulate a key definition, as follows:

Definition

Associated to any subfactor [math]A_0\subset A_1[/math] is the basic construction

[[math]] A_0\subset_eA_1\subset A_2 [[/math]]

with [math]A_2= \lt A_1,e \gt [/math] being the algebra generated by [math]A_1[/math] and by the Jones projection

[[math]] e:L^2(A_1)\to L^2(A_0) [[/math]]

acting on the Hilbert space [math]L^2(A_1)[/math].

The idea now, following as before Jones , will be that the inclusion [math]A_1\subset A_2[/math] appears as a kind of “reflection” of the original inclusion [math]A_0\subset A_1[/math], and also that the basic construction can be iterated, with all this leading to non-trivial results. We first have:

Proposition

Given a subfactor [math]A_0\subset A_1[/math] having finite index,

[[math]] [A_1:A_0] \lt \infty [[/math]]

the basic construction [math]A_0\subset_eA_1\subset A_2[/math] has the following properties:

• [math]A_2=JA_0'J[/math].
• [math]A_2=\overline{A_1+A_1eb}[/math].
• [math]A_2[/math] is a [math]{\rm II}_1[/math] factor.
• [math][A_2:A_1]=[A_1:A_0][/math].
• [math]eA_2e=A_0e[/math].
• [math]tr(e)=[A_1:A_0]^{-1}[/math].
• [math]tr(xe)=tr(x)[A_1:A_0]^{-1}[/math], for any [math]x\in A_1[/math].

Show Proof

All this is standard, by using the same type of mathematics as in the proof of Proposition 16.9, and for details here, we refer to Jones' paper .

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Let us perform now twice the basic construction, and see what we get. The result here, which is something more technical, at least at the first glance, is as follows:

Proposition

Associated to [math]A_0\subset A_1[/math] is the double basic construction

[[math]] A_0\subset_eA_1\subset_fA_2\subset A_3 [[/math]]

with [math]e:L^2(A_1)\to L^2(A_0)[/math] and [math]f:L^2(A_2)\to L^2(A_1)[/math] having the following properties:

[[math]] fef=[A_1:A_0]^{-1}f\quad,\quad efe=[A_1:A_0]^{-1}e [[/math]]

Show Proof

We have two formulae to be proved, the idea being as follows:

(1) The first formula in the statement is clear, because we have:

[[math]] fef =E(e)f =tr(e)f =[A_1:A_0]^{-1}f [[/math]]

(2) Regarding now the second formula, it is enough to check this on the dense subset [math](A_1+A_1eA_1)\Omega[/math]. Thus, we must show that for any [math]x,y,z\in A_1[/math], we have:

[[math]] efe(x+yez)\Omega=[A_1:A_0]^{-1}e(x+yez)\Omega [[/math]]

But this is something which is routine as well. See Jones .

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We can in fact perform the basic construction by recurrence, and we obtain:

Theorem

Associated to any subfactor [math]A_0\subset A_1[/math] is the Jones tower

[[math]] A_0\subset_{e_1}A_1\subset_{e_2}A_2\subset_{e_3}A_3\subset\ldots\ldots [[/math]]

with the Jones projections having the following properties:

• [math]e_i^2=e_i=e_i^*[/math].
• [math]e_ie_j=e_je_i[/math] for [math]|i-j|\geq2[/math].
• [math]e_ie_{i\pm1}e_i=[A_1:A_0]^{-1}e_i[/math].
• [math]tr(we_{n+1})=[A_1:A_0]^{-1}tr(w)[/math], for any word [math]w\in \lt e_1,\ldots,e_n \gt [/math].

Show Proof

This follows from Proposition 16.13 and Proposition 16.14, because the triple basic construction does not need in fact any further study. See .

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The relations found in Theorem 16.15 are in fact well-known, from the standard theory of the Temperley-Lieb algebra. This algebra, discovered by Temperley and Lieb in the context of statistical mechanics , has a very simple definition, as follows:

Definition

The Temperley-Lieb algebra of index [math]N\in[1,\infty)[/math] is defined as

[[math]] TL_N(k)=span(NC_2(k,k)) [[/math]]

with product given by vertical concatenation, with the rule

[[math]] \bigcirc=N [[/math]]

for the closed circles that might appear when concatenating.

In other words, the algebra [math]TL_N(k)[/math], depending on parameters [math]k\in\mathbb N[/math] and [math]N\in[1,\infty)[/math], is the linear span of the pairings [math]\pi\in NC_2(k,k)[/math]. The product operation is obtained by linearity, for the pairings which span [math]TL_N(k)[/math] this being the usual vertical concatenation, with the conventions that things go “from top to bottom”, and that each circle that might appear when concatenating is replaced by a scalar factor, equal to [math]N[/math].

In what concerns us, we will just need some elementary results. First, we have:

Proposition

The Temperley-Lieb algebra [math]TL_N(k)[/math] is generated by the diagrams

[[math]] \varepsilon_1={\ }^\cup_\cap\quad,\quad \varepsilon_2=|\!{\ }^\cup_\cap\quad,\quad \varepsilon_3=||\!{\ }^\cup_\cap\quad,\quad \ldots [[/math]]

which are all multiples of projections, in the sense that their rescaled versions

[[math]] e_i=N^{-1}\varepsilon_i [[/math]]

satisfy the abstract projection relations [math]e_i^2=e_i=e_i^*[/math].

Show Proof

We have two assertions here, the idea being as follows:

(1) The fact that the Temperley-Lieb algebra [math]TL_N(k)[/math] is indeed generated by the sequence [math]\varepsilon_1,\varepsilon_2,\ldots[/math] follows by drawing pictures, and more specifically by decomposing each basis element [math]\pi\in NC_2(k,k)[/math] as a product of such elements [math]\varepsilon_i[/math].

(2) Regarding now the projection assertion, when composing [math]\varepsilon_i[/math] with itself we obtain [math]\varepsilon_i[/math] itself, times a circle. Thus, according to our multiplication convention, we have:

[[math]] \varepsilon_i^2=N\varepsilon_i [[/math]]

Also, when turning upside-down [math]\varepsilon_i[/math], we obtain [math]\varepsilon_i[/math] itself. Thus, according to our involution convention for the Temperley-Lieb algebra, we have the following formula:

[[math]] \varepsilon_i^*=\varepsilon_i [[/math]]

We conclude that the rescalings [math]e_i=N^{-1}\varepsilon_i[/math] satisfy [math]e_i^2=e_i=e_i^*[/math], as desired.

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As a second result now, making the link with Theorem 16.15, we have:

Proposition

The standard generators [math]e_i=N^{-1}\varepsilon_i[/math] of the Temperley-Lieb algebra [math]TL_N(k)[/math] have the following properties, where [math]tr[/math] is the trace obtained by closing:

• [math]e_ie_j=e_je_i[/math] for [math]|i-j|\geq2[/math].
• [math]e_ie_{i\pm1}e_i=N^{-1}e_i[/math].
• [math]tr(we_{n+1})=N^{-1}tr(w)[/math], for any word [math]w\in \lt e_1,\ldots,e_n \gt [/math].

Show Proof

This follows indeed by doing some elementary computations with diagrams, in the spirit of those performed in the proof of Proposition 16.17.

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With the above results in hand, and still following Jones' paper , we can now reformulate Theorem 16.15 into something more conceptual, as follows:

Theorem

Given a subfactor [math]A_0\subset A_1[/math], construct its the Jones tower:

[[math]] A_0\subset_{e_1}A_1\subset_{e_2}A_2\subset_{e_3}A_3\subset\ldots\ldots [[/math]]

The rescaled sequence of projections [math]e_1,e_2,e_3,\ldots\in B(H)[/math] produces then a representation

[[math]] TL_N\subset B(H) [[/math]]

of the Temperley-Lieb algebra of index [math]N=[A_1:A_0][/math].

Show Proof

We know from Theorem 16.15 that the rescaled sequence of Jones projections [math]e_1,e_2,e_3,\ldots\in B(H)[/math] behaves algebrically exactly as the following [math]TL_N[/math] diagrams:

[[math]] \varepsilon_1={\ }^\cup_\cap\quad,\quad \varepsilon_2=|\!{\ }^\cup_\cap\quad,\quad \varepsilon_3=||\!{\ }^\cup_\cap\quad,\quad \ldots [[/math]]

But these diagrams generate [math]TL_N[/math], and so we have an embedding [math]TL_N\subset B(H)[/math], where [math]H[/math] is the Hilbert space where our subfactor [math]A_0\subset A_1[/math] lives, as claimed.

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Let us make the following key observation, also from :

Theorem

Given a finite index subfactor [math]A_0\subset A_1[/math], the graded algebra [math]P=(P_k)[/math] formed by the sequence of higher relative commutants

[[math]] P_k=A_0'\cap A_k [[/math]]

contains the copy of the Temperley-Lieb algebra constructed above, [math]TL_N\subset P[/math]. This graded algebra [math]P=(P_k)[/math] is called “planar algebra” of the subfactor.

Show Proof

As a first observation, since the Jones projection [math]e_1:A_1\to A_0[/math] commutes with [math]A_0[/math], we have [math]e_1\in P_2[/math]. By translation we obtain, for any [math]k\in\mathbb N[/math]:

[[math]] e_1,\ldots,e_{k-1}\in P_k [[/math]]

Thus we have indeed an inclusion of graded algebras [math]TL_N\subset P[/math], as claimed.

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As an interesting consequence of the above results, also from , we have:

Theorem

The index of subfactors [math]A\subset B[/math] is “quantized” in the [math][1,4][/math] range,

[[math]] N\in\left\{4\cos^2\left(\frac{\pi}{n}\right)\Big|n\geq3\right\}\cup[4,\infty] [[/math]]

with the obstruction coming from the existence of the representation [math]TL_N\subset B(H)[/math].

Show Proof

This comes from the basic construction, and more specifically from the combinatorics of the Jones projections [math]e_1,e_2,e_3,\ldots[/math], the idea being as folows:

(1) In order to best comment on what happens, when iterating the basic construction, let us record the first few values of the numbers in the statement:

[[math]] 4\cos^2\left(\frac{\pi}{3}\right)=1\quad,\quad 4\cos^2\left(\frac{\pi}{4}\right)=2 [[/math]]

[[math]] 4\cos^2\left(\frac{\pi}{5}\right)=\frac{3+\sqrt{5}}{2}\quad,\quad 4\cos^2\left(\frac{\pi}{6}\right)=3 [[/math]]

[[math]] \ldots [[/math]]

(2) When performing a basic construction, we obtain, by trace manipulations on [math]e_1[/math]:

[[math]] N\notin(1,2) [[/math]]

With a double basic construction, we obtain, by trace manipulations on [math] \lt e_1,e_2 \gt [/math]:

[[math]] N\notin\left(2,\frac{3+\sqrt{5}}{2}\right) [[/math]]

With a triple basic construction, we obtain, by trace manipulations on [math] \lt e_1,e_2,e_3 \gt [/math]:

[[math]] N\notin\left(\frac{3+\sqrt{5}}{2},3\right) [[/math]]

Thus, we are led to the conclusion in the statement, by a kind of recurrence, involving a certain family of orthogonal polynomials.

(3) In practice now, the most elegant way of proving the result is by using the fundamental fact, explained in Theorem 16.19, that that sequence of Jones projections [math]e_1,e_2,e_3,\ldots\subset B(H)[/math] generate a copy of the Temperley-Lieb algebra of index [math]N[/math]:

[[math]] TL_N\subset B(H) [[/math]]

With this result in hand, we must prove that such a representation cannot exist in index [math]N \lt 4[/math], unless we are in the following special situation:

[[math]] N=4\cos^2\left(\frac{\pi}{n}\right) [[/math]]

But this can be proved by using some suitable trace and positivity manipulations on [math]TL_N[/math], as in (2) above. For full details here, we refer to .

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So long for basic subfactor theory. As a continuation of the story, the subfactors of index [math]N\leq4[/math] are classified by the ADE graphs that we met in chapter 3. See .

General references

Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].