16b. Planar algebras

[math] \newcommand{\mathds}{\mathbb}[/math]

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Quite remarkably, the planar algebra structure of [math]TL_N[/math], taken in an intuitive sense, that of composing diagrams, in various possible ways, extends to a planar algebra structure on the sequence of higher relative commutants [math]P_n=A_0'\cap A_n[/math]. In order to discuss this, let us start with axioms for the planar algebras. Following Jones , we have:

Definition

The planar algebras are defined as follows:

• We consider rectangles in the plane, with the sides parallel to the coordinate axes, and taken up to planar isotopy, and we call such rectangles boxes.
• A labeled box is a box with [math]2n[/math] marked points on its boundary, [math]n[/math] on its upper side, and [math]n[/math] on its lower side, for some integer [math]n\in\mathbb N[/math].
• A tangle is labeled box, containing a number of labelled boxes, with all marked points, on the big and small boxes, being connected by noncrossing strings.
• A planar algebra is a sequence of finite dimensional vector spaces [math]P=(P_n)[/math], together with linear maps [math]P_{n_1}\otimes\ldots\otimes P_{n_k}\to P_n[/math], one for each tangle, such that the gluing of tangles corresponds to the composition of linear maps.

In this definition we are using rectangles, but everything being up to isotopy, we could have used instead circles with marked points, as in . Our choice for using rectangles comes from the main examples that we have in mind, to be discussed below, where the planar algebra structure is best viewed by using rectangles, as above.

Let us also mention that Definition 16.15 is something quite simplified, based on . As explained in , in order for subfactors to produce planar algebras and vice versa, there are quite a number of supplementary axioms that must be added, and in view of this, it is perhaps better to start with something stronger than Definition 16.15, as basic axioms. However, as before with rectangles vs circles, our axiomatic choices here are mainly motivated by the concrete examples that we have in mind. More on this later.

As a basic example of a planar algebra, we have the Temperley-Lieb algebra:

Theorem

The Temperley-Lieb algebra [math]TL_N[/math], viewed as graded algebra

[[math]] TL_N=(TL_N(n))_{n\in\mathbb N} [[/math]]

is a planar algebra, with the corresponding linear maps associated to the planar tangles

[[math]] TL_N(n_1)\otimes\ldots\otimes TL_N(n_k)\to TL_N(n) [[/math]]

appearing by putting the various [math]TL_N(n_i)[/math] diagrams into the small boxes of the given tangle, which produces a [math]TL_N(n)[/math] diagram.

Show Proof

This is something trivial, which follows from definitions:

(1) Assume indeed that we are given a planar tangle [math]\pi[/math], as in Definition 16.15, consisting of a box having [math]2n[/math] marked points on its boundary, and containing [math]k[/math] small boxes, having respectively [math]2n_1,\ldots,2n_k[/math] marked points on their boundaries, and then a total of [math]n+\Sigma n_i[/math] noncrossing strings, connecting the various [math]2n+\Sigma 2n_i[/math] marked points.

(2) We want to associate to this tangle [math]\pi[/math] a linear map as follows:

[[math]] T_\pi:TL_N(n_1)\otimes\ldots\otimes TL_N(n_k)\to TL_N(n) [[/math]]

For this purpose, by linearity, it is enough to construct elements as follows, for any choice of Temperley-Lieb diagrams [math]\sigma_i\in TL_N(n_i)[/math], with [math]i=1,\ldots,k[/math]:

[[math]] T_\pi(\sigma_1\otimes\ldots\otimes\sigma_k)\in TL_N(n) [[/math]]

(3) But constructing such an element is obvious, just by putting the various diagrams [math]\sigma_i\in TL_N(n_i)[/math] into the small boxes the given tangle [math]\pi[/math]. Indeed, this procedure produces a certain diagram in [math]TL_N(n)[/math], that we can call [math]T_\pi(\sigma_1\otimes\ldots\otimes\sigma_k)[/math], as above.

(4) Finally, we have to check that everything is well-defined up to planar isotopy, and that the gluing of tangles corresponds to the composition of linear maps. But both these checks are trivial, coming from the definition of [math]TL_N[/math], and we are done.

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As a conclusion to all this, [math]P=TL_N[/math] is indeed a planar algebra, but of somewhat “trivial” type, with the triviality coming from the fact that, in this case, the elements of [math]P[/math] are planar diagrams themselves, and so the planar structure appears trivially.

The Temperley-Lieb planar algebra [math]TL_N[/math] is however an important planar algebra, because it is the “smallest” one, appearing inside the planar algebra of any subfactor. But more on this later, when talking about planar algebras and subfactors.

Moving ahead now, here is our second basic example of a planar algebra, which is also “trivial” in the above sense, with the elements of the planar algebra being planar diagrams themselves, but which appears in a bit more complicated way:

Theorem

The Fuss-Catalan algebra [math]FC_{N,M}[/math], which appears by coloring the Temperley-Lieb diagrams with black/white colors, clockwise, as follows

[[math]] \circ\bullet\bullet\circ\circ\bullet\bullet\circ\ldots\ldots\ldots\circ\bullet\bullet\circ [[/math]]

and keeping those diagrams whose strings connect either [math]\circ-\circ[/math] or [math]\bullet-\bullet[/math], is a planar algebra, with again the corresponding linear maps associated to the planar tangles

[[math]] FC_{N,M}(n_1)\otimes\ldots\otimes FC_{N,M}(n_k)\to FC_{N,M}(n) [[/math]]

appearing by putting the various [math]FC_{N,M}(n_i)[/math] diagrams into the small boxes of the given tangle, which produces a [math]FC_{N,M}(n)[/math] diagram.

Show Proof

The proof here is nearly identical to the proof of Theorem 16.16, with the only change appearing at the level of the colors. To be more precise:

(1) Forgetting about upper and lower sequences of points, which must be joined by strings, a Temperley-Lieb diagram can be thought of as being a collection of strings, say black strings, which compose in the obvious way, with the rule that the value of the circle, which is now a black circle, is [math]N[/math]. And it is this obvious composition rule that gives the planar algebra structure, as explained in the proof of Theorem 16.16.

(2) Similarly, forgetting about points, a Fuss-Catalan diagram can be thought of as being a collection of strings, which come now in two colors, black and white. These Fuss-Catalan diagrams compose then in the obvious way, with the rule that the value of the black circle is [math]N[/math], and the value of the white circle is [math]M[/math]. And it is this obvious composition rule that gives the planar algebra structure, as before for [math]TL_N[/math].

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Getting back now to generalities, and to Definition 16.15, that of a general planar algebra, we have so far two illustrations for it, which, while both important, are both “trivial”, with the planar structure simply coming from the fact that, in both these cases, the elements of the planar algebra are planar diagrams themselves.

In general, the planar algebras can be more complicated than this, and we will see some further examples in a moment. However, the idea is very simple, namely “the elements of a planar algebra are not necessarily diagrams, but they behave like diagrams”.

In relation now with subfactors, the result, which extends Theorem 16.14, and which was found by Jones in , almost 20 years after , is as follows:

Theorem

Given a subfactor [math]A_0\subset A_1[/math], the collection [math]P=(P_n)[/math] of linear spaces

[[math]] P_n=A_0'\cap A_n [[/math]]

has a planar algebra structure, extending the planar algebra structure of [math]TL_N[/math].

Show Proof

We know from Theorem 16.14 that we have an inclusion as follows, coming from the basic construction, and with [math]TL_N[/math] itself being a planar algebra:

[[math]] TL_N\subset P [[/math]]

Thus, the whole point is that of proving that the trivial planar algebra structure of [math]TL_N[/math] extends into a planar algebra structure of [math]P[/math]. But this can be done via a long algebraic study, and for the full computation here, we refer to Jones' paper .

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As a first illustration for the above result, we have:

Theorem

We have the following universality results:

• The Temperley-Lieb algebra [math]TL_N[/math] appears inside the planar algebra of any subfactor [math]A_0\subset A_1[/math] having index [math]N[/math].
• The Fuss-Catalan algebra [math]FC_{N,M}[/math] appears inside the planar algebra of any subfactor [math]A_0\subset A_1[/math], in the presence of an intermediate subfactor [math]A_0\subset B\subset A_1[/math].

Show Proof

Here the first assertion is something that we already know, from Theorem 16.18, and the second assertion is something quite standard as well, by carefully working out the basic construction for [math]A_0\subset A_1[/math], in the presence of an intermediate subfactor [math]A_0\subset B\subset A_1[/math]. For details here, we refer to the paper of Bisch and Jones .

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As a free probability comment here, the Temperley-Lieb algebra, which appears by definition as the span of [math]NC_2[/math], is certainly a free probability object, and one way of being more concrete here is by saying that suitable fixed point subfactors associated to [math]S_N^+,O_N^+,U_N^+[/math] have planar algebra equal to [math]TL_N[/math]. See . As in what regards the Fuss-Catalan algebra, this is related to the bicolored partitions appearing in the study of [math]H_N^+[/math], and more generally of [math]H_N^{s+}[/math], and again, the precise subfactor statement about this concerns fixed point subfactors associated to the quantum groups [math]H_N^{s+}[/math]. See .

The above results raise the question on whether any planar algebra produces a subfactor. The answer here is yes, but with many subtleties, and in order to talk about this, we first need to introduce a certain distinguished [math]{\rm II}_1[/math] factor [math]R[/math], as follows:

Definition

The Murray-von Neumann hyperfinite [math]{\rm II}_1[/math] factor is

[[math]] R=\overline{\bigcup_iM_{n_i}(\mathbb C)}^{\,w} [[/math]]

independently of the choice of the algebras [math]M_{n_i}(\mathbb C)[/math], and of the embeddings between them.

To be more precise, all this is based on two theorems of Murray and von Neumann , stating on one hand that when performing the above inductive limit construction we obtain, after taking the weak closure, a certain [math]{\rm II}_1[/math] factor, and on the other hand, that the factor that we obtain is independent on the choice of the algebras [math]M_{n_i}(\mathbb C)[/math], and of the embeddings between them. All this is certainly non-trivial, and even less trivial is the following theorem, coming as a continuation of the work in , due to Connes :

Theorem

The Murray-von Neumann [math]{\rm II}_1[/math] factor [math]R[/math] is the unique [math]{\rm II}_1[/math] factor which is amenable, in the sense that we have a conditional expectation as follows:

[[math]] E:B(H)\to R [[/math]]

In particular, for a discrete group [math]\Gamma[/math] we have [math]L(\Gamma)=R[/math] precisely when [math]\Gamma\neq\{1\}[/math] has the infinite conjugacy class (ICC) property, and is amenable.

Show Proof

This is something fairly complicated, to the point of causing troubles not only to mathematicians, and no surprise here, but to physicists as well. In case you know a good physicist, the best is to ask that physicist, but there is no guarantee here, guy might well be clueless on all this. So, read from time to time operator algebras, say from , and once ready go through . And in the meantime do not hesitate to ask around, this being a good test for distinguishing good physicists from first-class physicists.

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Jokes left aside now, what is difficult in the above is the proof of “amenability implies hyperfiniteness”. Indeed, the converse can only be something standard, namely proving that a certain concrete algebra, [math]R[/math] from Definition 16.20, has a certain concrete property. As for the last assertion, this cannot be complicated either, because one of the possible definitions of the amenability of [math]\Gamma[/math] is in terms of an invariant mean [math]m:l^\infty(\Gamma)\to\mathbb C[/math], and this makes the connection with the expectation [math]E:B(l^2(\Gamma))\to L(\Gamma)[/math]. See .

Getting back now to subfactors, and to our questions regarding the correspondence between subfactors and planar algebras, these are difficult questions too, and the various answers to these questions can be summarized, a bit informally, as follows:

Theorem

We have the following results:

• Any planar algebra with positivity produces a subfactor.
• In particular, we have [math]TL[/math] and [math]FC[/math] type subfactors.
• In the amenable case, and with [math]A_1=R[/math], the correspondence is bijective.
• In general, we must take [math]A_1=L(F_\infty)[/math], and we do not have bijectivity.
• The axiomatization of [math]P[/math], in the case [math]A_1=R[/math], is not known.

Show Proof

All this is quite heavy, basically coming from the work of Popa in the 90s, using heavy functional analysis, the idea being as follows:

(1) As already mentioned after Definition 16.15, our planar algebra axioms here are something quite simplified, based on . However, when getting back to Theorem 16.18, the conclusion is that the subfactor planar algebras there satisfy a number of supplementary “positivity” conditions, basically coming from the positivity of the [math]{\rm II}_1[/math] factor trace. And the point is that, with these positivity conditions axiomatized, we reach to something which is equivalent to Popa's axiomatization of the lattice of higher relative commutants [math]A_i'\cap A_j[/math] of the finite index subfactors , obtained in the 90s via heavy functional analysis. For the full story here, and details, we refer to Jones' paper .

(2) The existence of the [math]TL_N[/math] subfactors, also known as “[math]A_\infty[/math] subfactors” in the literature, is something which was known for some time, since some early work of Popa on the subject. As for the existence of the [math]FC_{N,M}[/math] subfactors, this can be shown by using the intermediate subfactor picture, [math]A_0\subset B\subset A_1[/math], by composing two [math]A_\infty[/math] subfactors of suitable indices, [math]A_0\subset B[/math] and [math]B\subset A_1[/math]. For the full story here, we refer to , .

(3) This is something fairly heavy, as it is always the case with operator algebra results regarding hyperfiniteness and amenability, due to Popa , .

(4) This is something a bit more recent, obtained by further building on the above-mentioned constructions of Popa, and we refer here to and related work.

(5) This is the big open question in subfactors. The story here goes back to Jones' original paper , which contains at the end the question, due to Connes, of finding the possible values of the index for the irreducible subfactors of [math]R[/math]. This question, which certainly looks much easier than (5) in the statement, is in fact still open, now 40 years after its formulation, and with on one having any valuable idea in dealing with it.

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General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].