16a. Operator algebras

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

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We discuss here a question that we met several times in this book, in relation with our graph theory investigations, be them classical of quantum, namely the computation of the algebra generated by the adjacency matrix [math]d\in M_N(0,1)[/math] of a graph [math]X[/math], under the operations consisting in taking the spectral decomposition, and the color decomposition. This question makes in fact sense for any complex matrix, as follows: \begin{question} What is the algebra generated by a matrix [math]d\in M_N(\mathbb C)[/math],

[[math]] \lhd\, d\,\rhd=? [[/math]]

with respect to the spectral decomposition, and the color decomposition? \end{question} Here we use the above symbols in the lack of something known and standard, regarding this seemingly alien operation. However, as we will soon discover, there is nothing that alien regarding that operation, which is in fact something very familiar in operator algebras, knot theory, quantum field theory, and many more, namely the operation which consists in computing the associated planar algebra, in the sense of Jones .

So, this will be our plan, and the whole discussion will bring us on a long trip into modern mathematics, featuring advanced operator algebras, advanced linear algebra, and other advanced things, quite often with a flavor of modern topology and modern physics too, and as a bonus, we will reach to an answer to the following question too: \begin{question} What is the general mathematical theory behind the advanced topics that we saw in Parts I-II, namely walks on ADE graphs, and knot invariants? \end{question} With the answer to this latter question being, and you guessed right, again planar algebras in the sense of Jones . But probably enough advertisement, let us get to work. We will need some basic von Neumann algebra theory, coming as a complement to the basic [math]C^*[/math]-algebra theory developed in chapter 13, and we have here:

Theorem

For a [math]*[/math]-algebra of operators [math]A\subset B(H)[/math] the following conditions are equivalent, and if satisfied, we say that [math]A[/math] is a von Neumann algebra:

• [math]A[/math] is closed under the weak topology, making each [math]T\to Tx[/math] continuous.
• [math]A[/math] equals its bicommutant, [math]A=A''[/math], computed inside [math]B(H)[/math].

Show Proof

This is von Neumann's bicommutant theorem, that we actually invoked a few times already, when talking Tannakian duality, in its finite dimensional particular case, which is elementary, with the discussion here, and then the proof, being as follows:

(1) As a first comment, the weak topology on [math]B(H)[/math], making each [math]T\to Tx[/math] with [math]x\in H[/math] continuous, is indeed weaker than the norm topology, in the sense that we have:

[[math]] ||T_n-T||\to0\implies ||T_nx-Tx||\to0,\forall x\in H [[/math]]

In particular, we see that a von Neumann algebra in the sense of (1), that is, closed under the weak topology, must be a [math]C^*[/math]-algebra, that is, closed under the norm.

(2) Before getting further, let us see if the converse of this fact is true. This is certainly true in finite dimensions, [math]H=\mathbb C^N[/math], where we have [math]B(H)=M_N(\mathbb C)[/math], and where the operator [math]*[/math]-algebras [math]A\subset B(H)[/math] are as follows, automatically closed both for the norm topology, and the weak topology, and with these two topologies actually coinciding:

[[math]] A=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

(3) In infinite dimensions, however, things change. Indeed, let us first take a look at the most basic examples of commutative [math]C^*[/math]-algebras that we know, the commutative ones. These naturally appear from compact measured spaces [math]X[/math], as follows:

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

(4) But, it is pretty much clear that such an algebra will not be weakly closed, unless [math]X[/math] is discrete, with the details here being left to you. So, in infinite dimensions, there are far less von Neumann algebras than [math]C^*[/math]-algebras, with this being good to know.

(5) Still talking about this, the following natural question appears, what happens if we take the weak closure of the algebra [math]C(X)\subset B(L^2(X))[/math] considered above? And the answer here, obtained via some basic measure theory and functional analysis, that we will leave as an exercise, is that we obtain the following algebra:

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

(6) But this is quite interesting, because forgetting now about [math]C^*[/math]-algebras, what we have here is a nice method of producing von Neumann algebras, in the weakly closed sense, and with the measured space [math]X[/math] being no longer required to be compact.

(7) As a conclusion to all this, “von Neumann algebras have something to do with measure theory, in the same way as [math]C^*[/math]-algebras have something to do with topology”. Which sounds quite deep, so good, and time to stop here. More on this later.

(8) Hang on, we are not done yet with the preliminaries, because all the above was in relation with the condition (1) in the statement, and we still have the condition (2) in the statement to comment on. So, here we go again, with a basic exploration, of that condition. To start with, given a subalgebra [math]A\subset B(H)[/math], or even a subset [math]A\subset B(H)[/math], we can talk about its commutant inside [math]B(H)[/math], constructed as follows:

[[math]] A'=\left\{T\in B(H)\Big|TS=ST,\forall S\in A\right\} [[/math]]

Now if we take the commutant [math]A''[/math] of this commutant [math]A'[/math], it is obvious that the elements of the original algebra or set [math]A[/math] will be inside. Thus, we have an inclusion as follows:

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

(9) The question is now, why [math]A=A''[/math] should be equivalent to [math]A[/math] being weakly closed, and why should we care about this? These are both good questions, so let us start with the first one. As a first observation, in finite dimensions the bicommutant condition is automatic, because with [math]A\subset M_N(\mathbb C)[/math] being as in (2) above, its commutant is:

[[math]] A'=\mathbb C\oplus\ldots\oplus \mathbb C [[/math]]

But now, by taking again the commutant, we obtain the original algebra [math]A[/math]:

[[math]] A''=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

(10) Moving now to infinite dimensions, the first thought goes into taking the commutant of the basic examples of [math]C^*[/math]-algebras, [math]C(X)\subset B(L^2(X))[/math]. But here, up to some mesure theory and functional analysis work, that we will leave as an exercise, we are led to the following conclusion, which proves the bicommutant theorem in this case:

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

(11) Summarizing, we have some intuition on the condition [math]A=A''[/math] from the statement, and we can also say, based on the above, that the method for proving the bicommutant theorem would be that of establishing the following equality, for any [math]*[/math]-subalgebra [math]A\subset B(H)[/math], with on the right being the closure with respect to the weak topology:

[[math]] A''=\overline{A}^{\,w} [[/math]]

(12) Before getting to work, however, we still have a question to be answered, namely, why should we care about all this? I mean, the condition (1) in the statement, weak closedness, looks very nice and mathematical, that would be a good axiom for the von Neumann algebras, so why bothering with commutants, and with the condition (2).

(13) In answer, at the elementary level, and with my apologies for calling these damn things “elementary”, we have seen in chapters 12-13, when struggling with Tannakian duality, that the bicommutant operation and theorem can be something very useful.

(14) In answer too, at the advanced level now, in abstract quantum mechanics the vectors of the Hilbert space [math]x\in H[/math] are the states of the system, and the linear self-adjoint operators [math]T:H\to H[/math] are the observables, and taking the commutant of a set or algebra of observables is something extremely natural. And this is how von Neumann came upon such things, back in the 1930s, and looking now retrospectively, we can even say that his bicommutant theorem is not only important in the context of quantum mechanics, but even “makes abstract quantum mechanics properly work”. So, in short, trust me, with the present bicommutant theorem we are into first-class mathematics and physics.

(15) Time perhaps for the proof? We recall from (11) that we would like to prove the following equality, for any [math]*[/math]-algebra of operators [math]A\subset B(H)[/math]:

[[math]] A''=\overline{A}^{\,w} [[/math]]

(16) Let us first prove [math]\supset[/math]. Since we have [math]A\subset A''[/math], we just have to prove that [math]A''[/math] is weakly closed. But, assuming [math]T_i\to T[/math] weakly, we have indeed:

[[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]]

(17) Let us prove now [math]\subset[/math]. Here we must establish the following implication:

[[math]] T\in A''\implies T\in\overline{A}^{\,w} [[/math]]

For this purpose, we use an amplification trick. Consider indeed the Hilbert space [math]K[/math] obtained by summing [math]n[/math] times [math]H[/math] with itself:

[[math]] K=H\oplus\ldots\oplus H [[/math]]

The operators over [math]K[/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(K)[/math], as follows:

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

(18) The idea will be that of doing the computations in this 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 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_{1n}\\ \vdots&&\vdots\\ S_{n1}&\ldots&S_{nn} \end{pmatrix}\Big|S_{ij}\in A'\right\} [[/math]]

(19) We conclude from this that, given an operator [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:

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

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

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

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

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

Now since this holds for any vector [math]x\in K[/math], we conclude that any operator [math]T\in A''[/math] belongs to the weak closure of [math]A[/math]. Thus, we have [math]A''\subset\overline{A}^{\,w}[/math], as desired.

■

Very nice all this, but as you can see, the von Neumann algebras are far more subtle objects than the [math]C^*[/math]-algebras, and their proper understanding, even at the very basic level, is a far more complicated business than what we quickly did in chapter 13, for the [math]C^*[/math]-algebras. Welcome to the real quantum, the quantum mechanics one.

Moving ahead, the continuation of the story involves an accumulation of non-trivial results, due to Murray and von Neumann, from the 1930s and 1940s, and then due to Connes, much later, in the 1970s, the conclusions being as follows:

Theorem

The von Neumann algebras are as follows:

• In the commutative case, these are the algebras [math]A=L^\infty(X)[/math], with [math]X[/math] measured space, represented on [math]H=L^2(X)[/math], up to a multiplicity.
• If we write the center as [math]Z(A)=L^\infty(X)[/math], then we have a decomposition of type [math]A=\int_XA_x\,dx[/math], with the fibers [math]A_x[/math] having trivial center, [math]Z(A_x)=\mathbb C[/math].
• The factors, [math]Z(A)=\mathbb C[/math], can be fully classified in terms of [math]{\rm II}_1[/math] factors, which are those satisfying [math]\dim A=\infty[/math], and having a faithful trace [math]tr:A\to\mathbb C[/math].

Show Proof

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

(1) As already discussed above, it is clear that [math]L^\infty(X)[/math] is indeed a von Neumann algebra on [math]H=L^2(X)[/math]. The converse can be proved as well, by using spectral theory, one way of viewing this being by saying that, given a commutative von Neumann algebra [math]A\subset B(H)[/math], its elements [math]T\in A[/math] are commuting normal operators, so the Spectral Theorem for such operators applies, and gives [math]A=L^\infty(X)[/math], for some measured space [math]X[/math].

(2) This is von Neumann's reduction theory main result, whose statement is already quite hard to understand, and whose proof uses advanced functional analysis. To be more precise, in finite dimensions this is something that we know well, with the formula [math]A=\int_XA_x\,dx[/math] corresponding to our usual direct sum decomposition, namely:

[[math]] A=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

In infinite dimensions, things are more complicated, but the idea remains the same, namely using (1) for the commutative von Neumann algebra [math]Z(A)[/math], as to get a measured space [math]X[/math], and then making your way towards a decomposition of type [math]A=\int_XA_x\,dx[/math].

(3) This is something fairly heavy, due to Murray-von Neumann and Connes, the idea being that the other factors can be basically obtained via crossed product constructions. To be more precise, the various type of factors can be classified as follows:

-- Type I. These are the matrix algebras [math]M_N(\mathbb C)[/math], called of type [math]{\rm I}_N[/math], and their infinite generalization, [math]B(H)[/math] with [math]H[/math] infinite dimensional, called of type [math]{\rm I}_\infty[/math]. Although these factors are very interesting and difficult mathematical objects, from the perspective of the general von Neumann algebra classification work, they are dismissed as “trivial”.

-- Type II. These are the infinite dimensional factors having a trace, which is a usual trace [math]tr:A\to\mathbb C[/math] in the type [math]{\rm II}_1[/math] case, and is something more technical, possibly infinite, in the remaining case, the type [math]{\rm II}_\infty[/math] one, with these latter factors being of the form [math]B(H)\otimes A[/math], with [math]A[/math] being a [math]{\rm II}_1[/math] factor, and with [math]H[/math] being an infinite dimensional Hilbert space.

-- Type III. These are the factors which are infinite dimensional, and do not have a trace [math]tr:A\to\mathbb C[/math]. Murray and von Neumann struggled a lot with such beasts, with even giving an example being a non-trivial task, but later Connes came and classified them, basically showing that they appear from [math]{\rm II}_1[/math] factors, via crossed product constructions.

■

So long for basic, or rather advanced but foundational, von Neumann algebra theory. In what follows we will focus on the [math]{\rm II}_1[/math] factors, according to the following principle:

\begin{principle} The building blocks of the von Neumann algebra theory are the [math]{\rm II}_1[/math] factors, which are the von Neumann algebras having the following properties:

• They are infinite dimensional, [math]\dim A=\infty[/math].
• They are factors, their center being [math]Z(A)=\mathbb C[/math].
• They have a faithful trace [math]tr:A\to\mathbb C[/math].

\end{principle} But you might perhaps ask, is it even clear that such beasts exist? Good point, and in answer, given a discrete group [math]\Gamma[/math], you can talk about its von Neumann algebra, obtained by talking the weak closure of the usual group algebra, or group [math]C^*[/math]-algebra:

[[math]] L(\Gamma)\subset B(l^2(\Gamma)) [[/math]]

This algebra is then infinite dimensional when [math]\Gamma[/math] is infinite, and also has a trace, given on group elements by [math]tr(g)=\delta_{g1}[/math]. As for the center, this consists of the functions on [math]\Gamma[/math] which are constant on the conjugacy classes, so when [math]\Gamma[/math] has infinite conjugacy classes, called ICC property, what we have is a factor. Thus, as a conclusion, when [math]\Gamma[/math] is infinite and has the ICC property, its von Neumann algebra [math]L(\Gamma)[/math] is a [math]{\rm II}_1[/math] factor.

Let us summarize this finding, along with a bit more, as follows:

Theorem

We have the following examples of [math]{\rm II}_1[/math] factors:

• The group von Neumann algebras [math]L(\Gamma)[/math], with [math]\Gamma[/math] being an infinite group, having the infinite conjugacy class (ICC) property.
• The Murray-von Neumann hyperfinite factor [math]R=\overline{\cup_kM_{n_k}(\mathbb C)}^{\,w}[/math], with the limit being independent on the summands, and on the inclusions between them.
• With the remark that when [math]\Gamma[/math] as above is assumed to be amenable, its associated [math]{\rm II}_1[/math] factor [math]L(\Gamma)[/math] is the Murray-von Neumann hyperfinite factor [math]R[/math].

Show Proof

Here the first assertion comes from the above discussion, and the rest, regarding the factor [math]R[/math], is due to Murray and von Neumann, using standard functional analysis. With the remark however that the notion of hyperfiniteness can be plugged into the general considerations from Theorem 16.4, and with the resulting questions, which are of remarkable difficulty, having been solved only relatively recently, basically by Connes in the 1970s, and with a last contribution by Haagerup in the 1980s, the general idea being that, in the end, everything hyperfinite can be reconstructed from [math]R[/math].

■

Many other things can be said, along these lines, and if truly interested in theoretical physics, be that quantum mechanics, or statistical mechanics, or other, have a look at all this, von Neumann algebras, this is first-class mathematical technology.

General references

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