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We have now a quite complete picture of free probability from a combinatorial point of view, in relation with basic questions from random matrices and quantum groups. In this final chapter we go for the real thing, namely discussing the connections between free probability and selected topics from von Neumann algebra theory.

We already know a few things about the algebras of operators [math]A\subset B(H)[/math] which are norm closed. The von Neumann algebras will be by definition those such algebras which are weakly closed. In order to discuss this, let us start with a standard result:

Proposition

For an algebra [math]A\subset B(H)[/math], the following are equivalent:

• [math]A[/math] is closed under the weak operator topology, making each of the linear maps [math]T\to \lt Tx,y \gt [/math] continuous.
• [math]A[/math] is closed under the strong operator topology, making each of the linear maps [math]T\to Tx[/math] continuous.

In the case where these conditions are satisfied, [math]A[/math] is closed under the norm topology.

Show Proof

There are several statements here, the proof being as follows:

(1) It is clear that the norm topology is stronger than the strong operator topology, which is in turn stronger than the weak operator topology. Thus, we are left with proving that for any operator algebra [math]A\subset B(H)[/math], strongly closed implies weakly closed.

(2) But this latter fact is standard, and can be proved by using an amplification trick. Consider the Hilbert space obtained by summing [math]k[/math] times [math]H[/math] with itself:

[[math]] H^+=H\oplus\ldots\oplus H [[/math]]

The operators over [math]H^+[/math] can be regarded as being square matrices with entries in [math]B(H)[/math], and in particular, we have a representation [math]\pi:B(H)\to B(H^+)[/math], given by:

[[math]] \pi(T)=\begin{pmatrix} T\\ &\ddots\\ &&T \end{pmatrix} [[/math]]

Assume now that we are given an operator [math]T\in\bar{A}[/math], with the bar denoting the weak closure. We have, by using the Hahn-Banach theorem, for any [math]\xi\in H^+[/math]:

[[math]] \begin{eqnarray*} T\in\bar{A} &\implies&\pi(T)\in\overline{\pi(A)}\\ &\implies&\pi(T)x\in\overline{\pi(A)\xi}\\ &\implies&\pi(T)x\in\overline{\pi(A)\xi}^{\,||.||} \end{eqnarray*} [[/math]]

Now observe that the last formula tells us that for any [math]\xi=(\xi_1,\ldots,\xi_k)[/math], and any [math]\varepsilon \gt 0[/math], we can find an operator [math]S\in A[/math] such that the following holds, for any [math]i[/math]:

[[math]] ||S\xi_i-T\xi_i|| \lt \varepsilon [[/math]]

It follows that [math]T[/math] belongs to the strong operator closure of [math]A[/math], as desired.

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In the above statement the terminology, while quite standard, is a bit confusing, because the norm topology is stronger than the strong operator topology. As a solution to this issue, we agree in what follows to call the norm topology “strong”, and the weak and strong operator topologies “weak”, whenever these two topologies coincide.

With this convention, the operator algebras [math]A\subset B(H)[/math] from Proposition 16.1 are those which are weakly closed, and we can now formulate:

Definition

A von Neumann algebra is a [math]*[/math]-algebra of operators

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

which is closed under the weak topology.

As basic examples, we have the algebra [math]B(H)[/math] itself, then the singly generated von Neumann algebras, [math]A= \lt T \gt [/math], with [math]T\in B(H)[/math], and then the multiply generated von Neumann algebras, namely [math]A= \lt T_i \gt [/math], with [math]T_i\in B(H)[/math]. There are many other examples, and also general methods for constructing examples, and we will discuss this later.

At the level of the general results, we first have the bicommutant theorem of von Neumann, which provides a useful alternative to Definition 16.2, as follows:

Theorem

For a [math]*[/math]-algebra [math]A\subset B(H)[/math], the following are equivalent:

• [math]A[/math] is weakly closed, so it is a von Neumann algebra.
• [math]A[/math] equals its algebraic bicommutant [math]A''[/math], taken inside [math]B(H)[/math].

Show Proof

Since the commutants are weakly closed, it is enough to show that weakly closed implies [math]A=A''[/math]. For this purpose, we will prove something a bit more general, stating that given a [math]*[/math]-algebra of operators [math]A\subset B(H)[/math], the following holds, with [math]A''[/math] being the bicommutant inside [math]B(H)[/math], and with [math]\bar{A}[/math] being the weak closure:

[[math]] A''=\bar{A} [[/math]]

We can prove this by double inclusion, as follows:

“[math]\supset[/math]” Since any operator commutes with the operators that it commutes with, we have an inclusion [math]E\subset E''[/math], valid for any set [math]E\subset B(H)[/math]. In particular, we have:

[[math]] A\subset A'' [[/math]]

Our claim now is that the algebra [math]A''\subset B(H)[/math] is closed, with respect to the strong operator topology. Indeed, assuming that we have [math]T_i\to T[/math] in this topology, we have:

[[math]] \begin{eqnarray*} T_i\in A'' &\implies&ST_i=T_iS,\ \forall S\in A'\\ &\implies&ST=TS,\ \forall S\in A'\\ &\implies&T\in A \end{eqnarray*} [[/math]]

Thus our claim is proved, and together with Proposition 16.1, which allows to pass from the strong to the weak operator topology, this gives the desired inclusion, namely:

[[math]] \bar{A}\subset A'' [[/math]]

“[math]\subset[/math]” Here we must prove that we have the following implication, valid for any operator [math]T\in B(H)[/math], with the bar denoting as usual the weak operator closure:

[[math]] T\in A''\implies T\in\bar{A} [[/math]]

For this purpose, we use the same amplification trick as in the proof of Proposition 16.1. Consider the Hilbert space obtained by summing [math]k[/math] times [math]H[/math] with itself:

[[math]] H^+=H\oplus\ldots\oplus H [[/math]]

The operators over [math]H^+[/math] can be regarded as being square matrices with entries in [math]B(H)[/math], and in particular, we have a representation [math]\pi:B(H)\to B(H^+)[/math], given by:

[[math]] \pi(T)=\begin{pmatrix} T\\ &\ddots\\ &&T \end{pmatrix} [[/math]]

The idea will be that of doing the computations in this latter representation. First, in this representation, the image of our algebra [math]A\subset B(H)[/math] is given by:

[[math]] \pi(A)=\left\{\begin{pmatrix} T\\ &\ddots\\ &&T \end{pmatrix}\Big|T\in A\right\} [[/math]]

We can now compute the commutant of this image, exactly as in the usual scalar matrix case, and we obtain the following formula:

[[math]] \pi(A)'=\left\{\begin{pmatrix} S_{11}&\ldots&S_{1k}\\ \vdots&&\vdots\\ S_{k1}&\ldots&S_{kk} \end{pmatrix}\Big|S_{ij}\in A'\right\} [[/math]]

We conclude from this that, given [math]T\in A''[/math] as above, we have:

[[math]] \begin{pmatrix} T\\ &\ddots\\ &&T \end{pmatrix}\in\pi(A)'' [[/math]]

In other words, the conclusion of all this is that we have the following implication:

[[math]] T\in A''\implies \pi(T)\in\pi(A)'' [[/math]]

Now given [math]\xi\in H^+[/math], consider the orthogonal projection [math]P\in B(H^+)[/math] on the norm closure of the vector space [math]\pi(A)\xi\subset H^+[/math]. Since the subspace [math]\pi(A)\xi\subset H^+[/math] is invariant under the action of [math]\pi(A)[/math], so is its norm closure inside [math]H^+[/math], and we obtain from this:

[[math]] P\in\pi(A)' [[/math]]

By combining this with what we found above, we conclude that:

[[math]] T\in A''\implies \pi(T)P=P\pi(T) [[/math]]

Now since this holds for any [math]\xi\in H^+[/math], it follows that any [math]T\in A''[/math] belongs to the strong operator closure of [math]A[/math]. By using now Proposition 16.1, which allows us to pass from the strong to the weak operator closure, we conclude that we have [math]A''\subset\bar{A}[/math], as desired.

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In order to develop now some general theory for the von Neumann algebras, let us start by investigating the commutative case. The result here is as follows:

Theorem

The commutative von Neumann algebras are the algebras of type

[[math]] A=L^\infty(X) [[/math]]

with [math]X[/math] being a measured space.

Show Proof

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

(1) In one sense, we must prove that given a measured space [math]X[/math], we can realize the commutative algebra [math]A=L^\infty(X)[/math] as a von Neumann algebra, on a certain Hilbert space [math]H[/math]. But this can be done as follows, using a probability measure on [math]X[/math]:

[[math]] L^\infty(X)\subset B(L^2(X))\quad,\quad f\to(g\to fg) [[/math]]

(2) In the other sense, given a commutative von Neumann algebra [math]A\subset B(H)[/math], any operator [math]T\in A[/math] is normal. So, ley us pick a linear space basis [math]\{T_i\}\subset A[/math], as to have:

[[math]] A= \lt T_i \gt [[/math]]

The generators [math]T_i\in B(H)[/math] are then commuting normal operators, and by using the spectral theorem for such families of operators, we obtain the result.

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The above result is very interesting, because it shows that an arbitrary von Neumann algebra [math]A\subset B(H)[/math] can be thought of as being of the form [math]A=L^\infty(X)[/math], with [math]X[/math] being a “quantum measured space”. Thus, we have here a connection with the various quantum group and noncommutative geometry considerations made before.

Moving ahead now, we will be interested here in the “free” von Neumann algebras. These algebras, traditionally called factors, can be axiomatized as follows:

Definition

A factor is a von Neumann algebra [math]A\subset B(H)[/math] whose center

[[math]] Z(A)=A\cap A' [[/math]]

which is a commutative von Neumann algebra, reduces to the scalars, [math]Z(A)=\mathbb C[/math].

Here the fact that the center is indeed a von Neumann algebra follows from the bicommutant theorem, which shows that the commutant of any [math]*[/math]-algebra is a von Neumann algebra. Thus, the intersection [math]Z(A)=A\cap A'[/math] is indeed a von Neumann algebra.

Before going further, let us mention that, besides their intuitive freeness, there are some deeper reasons too for the consideration of factors, which among others fully justify the term “factor”, coming from the following advanced theorem of von Neumann:

Theorem

Given a von Neumann algebra [math]A\subset B(H)[/math], if we write its center as

[[math]] Z(A)=L^\infty(X) [[/math]]

then we have a decomposition as follows, with the fibers [math]A_x[/math] being factors:

[[math]] A=\int_XA_x\,dx [[/math]]

Moreover, in the case where [math]A[/math] has a trace, [math]tr:A\to\mathbb C[/math], this trace decomposes as

[[math]] tr=\int_Xtr_x\,dx [[/math]]

with each [math]tr_x:A_x\to\mathbb C[/math] being the restriction of [math]tr[/math] to the factor [math]A_x[/math].

Show Proof

As a first observation, this is something that we know to hold in finite dimensions, because here the algebra decomposes as follows, with the summands corresponding precisely to the points of the spectrum of the center, [math]Z(A)\simeq\mathbb C^k[/math]:

[[math]] A=M_{N_1}(\mathbb C)\oplus\ldots\oplus M_{N_k}(\mathbb C) [[/math]]

In general, however, this is something quite difficult to prove, requiring a good knowledge of advanced operator theory and functional analysis. We will not really need this result in what follows, and we refer here to any good operator algebra book.

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Moving ahead now, in order to do probability on our factors we will need a trace as well. Leaving aside the somewhat trivial case [math]A=M_N(\mathbb C)[/math], we are led in this way to:

Definition

A [math]{\rm II}_1[/math] factor is a von Neumann algebra [math]A\subset B(H)[/math] which is infinite dimensional, has trivial center, and has a trace [math]tr:A\to\mathbb C[/math].

As a first observation, according to Theorem 16.6, such factors are exactly those appearing in the spectral decomposition of the von Neumann algebras [math]A\subset B(H)[/math] which have traces, [math]tr:A\to\mathbb C[/math], provided that we add some extra axioms which avoid trivial summands of type [math]M_N(\mathbb C)[/math]. Moreover, by results of Connes, adding to those of von Neumann, and which are non-trivial as well, the non-tracial case basically reduces to the tracial case, via certain crossed product type operations, and so the conclusion is that “the [math]{\rm II}_1[/math] factors are the building blocks of the von Neumann algebra theory”.

Summarizing, some heavy things going on here. In what follows we will be mainly interested in concrete mathematics and combinatorics, and we will take Definition 16.7 as it is, as a simple and intuitive definition for the “free von Neumann algebras”.

There are many things that can be said about the [math]{\rm II}_1[/math] factors, and of particular interest is the following key result of Murray and von Neumann , which clarifies the situation with the various Hilbert space representations [math]A\subset B(H)[/math] of a given [math]{\rm II}_1[/math] factor [math]A[/math]:

Theorem

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

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

which for the standard form, where [math]H=L^2(A)[/math], takes the value [math]1[/math], and which in general mesures how far is [math]A\subset B(H)[/math] from the standard form.

Show Proof

There are several proofs for this fact, the idea being as follows:

(1) We can amplify the standard representation of [math]A[/math], on the Hilbert space [math]L^2(A)[/math], into a representation on [math]L^2(A)\otimes l^2(\mathbb N)[/math], and then cut it down with a projection. We obtain in this way a whole family of embeddings [math]A\subset B(H)[/math], which are quite explicit.

(2) The point now is that of proving, via a technical [math]2\times2[/math] matrix trick, that any representation [math]A\subset B(H)[/math] appears in this way. In this picture, the coupling constant appears as the trace of the projection used to cut down [math]L^2(A)\otimes l^2(\mathbb N)[/math].

(3) Thus, we are led to the conclusion in the statement. Alternatively, the coupling constant can be defined as follows, with the number on the right being independent of the choice on a nonzero vector [math]x\in H[/math], and with this being the original definition from :

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

We refer to , or for instance to the book , for more details here.

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Following Jones , given a [math]{\rm II}_1[/math] factor [math]A_0[/math], let us discuss now the representations [math]A_0\subset A_1[/math], with [math]A_1[/math] being another [math]{\rm II}_1[/math] factor. This is a quite natural notion too, and perhaps even more natural than the representations [math]A_0\subset B(H)[/math], because we have decided in the above that the [math]{\rm II}_1[/math] factors [math]A_1[/math], and not the full operator algebras [math]B(H)[/math], are the correct infinite dimensional generalization of the usual matrix algebras [math]M_N(\mathbb C)[/math].

Given an inclusion of [math]{\rm II}_1[/math] factors [math]A_0\subset A_1[/math], a first question is that of defining its index, measuring how big is [math]A_1[/math], when compared to [math]A_0[/math]. This can be done as follows:

Theorem

Given an inclusion of [math]{\rm II}_1[/math] factors [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 standard, with the fact that the index as defined by the above formula is indeed independent of the ambient Hilbert space [math]H[/math] coming from the various basic properties of the coupling constant, from Theorem 16.8 and its proof.

■

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:

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 standard projection

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

also called Jones projection, acting on the Hilbert space [math]L^2(A_1)[/math].

The idea now, following , will be that [math]A_1\subset A_2[/math] appears as a kind of “reflection” of [math]A_0\subset A_1[/math], and also that the basic construction can be iterated, and with all this leading to non-trivial results. Let us start by further studying the basic construction:

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

This is routine, with [math]J(T)=T^*[/math], and we refer here to Jones .

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The above result is quite interesting, potentially leading to some interesting mathematics, so 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

Again, this is standard, and for details, we refer to 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 16.11 and Proposition 16.12, because the triple basic construction does not need in fact any further study. See .

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The point now is that the relations found in Theorem 16.13 are well-known, from the standard theory of the Temperley-Lieb algebra . Thus, still following Jones' paper , we can now reformulate Theorem 16.13 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 16.13 that the rescaled sequence of 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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General references

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