14c. Von Neumann algebras

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

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Let us discuss now some applications of the construction [math]H\to G[/math], to questions from mathematical physics. We will need some basic von Neumann algebra theory, coming as a complement to the basic [math]C^*[/math]-algebra theory from chapter 13, as follows:

Theorem

The von Neumann algebras, which are the [math]*[/math]-algebras of operators

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

closed under the weak operator topology, making each [math]T\to Tx[/math] continuous, are as follows:

• They are exactly the [math]*[/math]-algebras of operators [math]A\subset B(H)[/math] which are equal to their bicommutant, [math]A=A''[/math].
• 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].
• The [math]{\rm II}_1[/math] factors enjoy the “continuous dimension geometry” property, in the sense that the traces of their projections can take any values in [math][0,1][/math].
• Among the [math]{\rm II}_1[/math] factors, the most important one is the Murray-von Neumann hyperfinite factor [math]R[/math], obtained as an inductive limit of matrix algebras.

Show Proof

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

(1) This is von Neumann's bicommutant theorem, which is well-known in finite dimensions, and whose proof in general is not that complicated, either.

(2) It is clear, via basic measure theory, 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.

(3) This is von Neumann's reduction theory main result, whose statement is already quite hard to understand, and whose proof uses advanced functional analysis.

(4) This is something heavy, due to Murray-von Neumann and Connes, the idea being that the other factors can be basically obtained via crossed product constructions.

(5) This is a gem of functional analysis, with the rational traces being relatively easy to obtain, and with the irrational ones coming from limiting arguments.

(6) Once again, heavy results, by Murray-von Neumann and Connes, the idea being that any finite dimensional construction always leads to the same factor, called [math]R[/math].

■

In relation now with our questions, variations of von Neumann's reduction theory idea, basically using the abelian subalgebra [math]Z(A)\subset A[/math], include the use of maximal abelian subalgebras [math]B\subset A[/math], called MASA. In the finite von Neumann algebra case, where we have a trace, the use of orthogonal MASA is a standard method as well, and we have:

Definition

A pair of orthogonal MASA inside a von Neumann algebra [math]A[/math] with a trace, [math]tr:A\to\mathbb C[/math], is a pair of maximal abelian subalgebras

[[math]] B,C\subset A [[/math]]

which are orthogonal with respect to the trace, in the sense that we have

[[math]] (B\ominus\mathbb C1)\perp(C\ominus\mathbb C1) [[/math]]

with the scalar product being by definition given by [math] \lt b,c \gt =tr(bc^*)[/math].

Observe that, by taking into account the multiples of the identity, the orthogonality condition appearing above reformulates as follows:

[[math]] tr(bc)=tr(b)tr(c) [[/math]]

The above notion is potentially useful in the infinite dimensional context, in relation with various structure and classification problems for the [math]{\rm II}_1[/math] factors. However, as a toy example, we can try and see what happens for the simplest factor that we know, namely the matrix algebra [math]M_N(\mathbb C)[/math], with its usual trace. In this context, we have the following surprising observation of Popa , making the link with the Hadamard matrices:

Theorem

Up to a conjugation by a unitary, the pairs of orthogonal MASA in the simplest factor, namely the matrix algebra [math]M_N(\mathbb C)[/math], are as follows,

[[math]] A=\Delta\quad,\quad B=H\Delta H^* [[/math]]

with [math]\Delta\subset M_N(\mathbb C)[/math] being the diagonal matrices, and with [math]H\in M_N(\mathbb C)[/math] being Hadamard.

Show Proof

Any MASA in [math]M_N(\mathbb C)[/math] being conjugated to the diagonal algebra [math]\Delta[/math], we can assume, up to conjugation by a unitary, that we have, for a certain [math]U\in U_N[/math]:

[[math]] A=\Delta\quad,\quad B=U\Delta U^* [[/math]]

Now observe that given two diagonal matrices [math]D,E\in\Delta[/math], we have:

[[math]] \begin{eqnarray*} tr(D\cdot UEU^*) &=&\frac{1}{N}\sum_i(DUEU^*)_{ii}\\ &=&\frac{1}{N}\sum_{ij}D_{ii}U_{ij}E_{jj}\bar{U}_{ij}\\ &=&\frac{1}{N}\sum_{ij}D_{ii}E_{jj}|U_{ij}|^2 \end{eqnarray*} [[/math]]

Thus, the orthogonality condition [math]A\perp B[/math] reformulates as follows:

[[math]] \frac{1}{N}\sum_{ij}D_{ii}E_{jj}|U_{ij}|^2=\frac{1}{N^2}\sum_{ij}D_{ii}E_{jj} [[/math]]

But this tells us precisely that the entries [math]|U_{ij}|[/math] must have the same absolute value:

[[math]] |U_{ij}|=\frac{1}{\sqrt{N}} [[/math]]

Thus the rescaled matrix [math]H=\sqrt{N}U[/math] must be Hadamard, as desired.

■

Along the same lines, but at a more advanced level, we have the following result:

Theorem

Given a complex Hadamard matrix [math]H\in M_N(\mathbb C)[/math], the diagram formed by the associated pair of orthogonal MASA, namely

[[math]] \xymatrix@R=35pt@C35pt{ \Delta\ar[r]&M_N(\mathbb C)\\ \mathbb C\ar[u]\ar[r]&H\Delta H^*\ar[u] } [[/math]]

is a commuting square in the sense of subfactor theory, in the sense that the expectations onto [math]\Delta,H\Delta H^*[/math] commute, and their product is the expectation onto [math]\mathbb C[/math].

Show Proof

It follows from definitions that the expectation [math]E_\Delta:M_N(\mathbb C)\to\Delta[/math] is the operation which consists in keeping the diagonal, and erasing the rest:

[[math]] M\to M_\Delta [[/math]]

Consider now the other expectation, namely:

[[math]] E_{H\Delta H^*}:M_N(\mathbb C)\to H\Delta H^* [[/math]]

It is better to identify this with the following expectation, with [math]U=H/\sqrt{N}[/math]:

[[math]] E_{U\Delta U^*}:M_N(\mathbb C)\to U\Delta U^* [[/math]]

This latter expectation must be given by a formula of type [math]M\to UX_\Delta U^*[/math], with [math]X[/math] satisfying the following condition:

[[math]] \lt M,UDU^* \gt = \lt UX_\Delta U^*,UDU^* \gt \quad,\quad\forall D\in\Delta [[/math]]

The scalar products being given by [math] \lt a,b \gt =tr(ab^*)[/math], this condition reads:

[[math]] tr(MUD^*U^*)=tr(X_\Delta D^*)\quad,\quad\forall D\in\Delta [[/math]]

Thus [math]X=U^*MU[/math], and the formulae of our two expectations are as follows:

[[math]] \begin{eqnarray*} E_\Delta(M)&=&M_\Delta\\ E_{U\Delta U^*}(M)&=&U(U^*MU)_\Delta U^* \end{eqnarray*} [[/math]]

With these formulae in hand, we have the following computation:

[[math]] \begin{eqnarray*} (E_\Delta E_{U\Delta U^*}M)_{ij} &=&\delta_{ij}(U(U^*MU)_\Delta U^*)_{ii}\\ &=&\delta_{ij}\sum_kU_{ik}(U^*MU)_{kk}\bar{U}_{ik}\\ &=&\delta_{ij}\sum_k\frac{1}{N}\cdot(U^*MU)_{kk}\\ &=&\delta_{ij}tr(U^*MU)\\ &=&\delta_{ij}tr(M)\\ &=&(E_\mathbb CM)_{ij} \end{eqnarray*} [[/math]]

As for the other composition, the computation here is similar, as follows:

[[math]] \begin{eqnarray*} (E_{U\Delta U^*}E_\Delta M)_{ij} &=&(U(U^*M_\Delta U)_\Delta U^*)_{ij}\\ &=&\sum_kU_{ik}(U^*M_\Delta U)_{kk}\bar{U}_{jk}\\ &=&\sum_{kl}U_{ik}\bar{U}_{lk}M_{ll}U_{lk}\bar{U}_{jk}\\ &=&\frac{1}{N}\sum_{kl}U_{ik}M_{ll}\bar{U}_{jk}\\ &=&\delta_{ij}tr(M)\\ &=&(E_\mathbb CM)_{ij} \end{eqnarray*} [[/math]]

Thus, we have indeed a commuting square, as claimed.

■

As a conclusion, all this leads us into commuting squares and subfactor theory. So, let us explain now the basic theory here. As a first object, which will be central in what follows, we have the Temperley-Lieb algebra , constructed 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 formal linear span of the noncrossing 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]. Observe that there is a connection here with [math]S_N^+[/math], and more specifically with the category of noncrossing partitions [math]NC[/math] producing [math]S_N^+[/math], due to the following fact:

Proposition

We have bijections

[[math]] NC(k)\simeq NC_2(2k)\simeq NC_2(k,k) [[/math]]

constructed by fattening/shrinking and rotating/flattening, as follows:

• The application [math]NC(k)\to NC_2(2k)[/math] is the “fattening” one, obtained by doubling all the legs, and doubling all the strings as well.
• Its inverse [math]NC_2(2k)\to NC(k)[/math] is the “shrinking” application, obtained by collapsing pairs of consecutive neighbors.
• The bijection [math]NC_2(2k)\simeq NC_2(k,k)[/math] is obtained by rotating and flattening the noncrossing pairings, in the obvious way.

Show Proof

The fact that the two operations in (1,2) are indeed inverse to each other is clear, by computing the corresponding two compositions, with the remark that the construction of the fattening operation requires indeed the partitions to be noncrossing. Thus, we are led to the conclusions in the statement.

■

Getting back now to von Neumann algebras, following Jones , consider an inclusion of [math]{\rm II}_1[/math] factors, which is actually something quite natural in quantum physics:

[[math]] A_0\subset A_1 [[/math]]

We can consider the orthogonal projection [math]e_1:A_1\to A_0[/math], and set:

[[math]] A_2= \lt A_1,e_1 \gt [[/math]]

This procedure, discovered by Jones and called “basic construction”, can be iterated, and we obtain in this way a whole tower of [math]{\rm II}_1[/math] factors, as follows:

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

The basic construction is something quite subtle, making deep connections with advanced mathematics and physics. All this was discovered by Jones in the early 80s, and his main result from , which came as a big surprise at that time, along with some supplementary fundamental work, done later, in , can be summarized as follows:

Theorem

Let [math]A_0\subset A_1[/math] be an inclusion of [math]{\rm II}_1[/math] factors.

• The sequence of Jones projections [math]e_1,e_2,e_3,\ldots\in B(H)[/math] produces a Hilbert space representation of the Temperley-Lieb algebra
  [[math]] TL_N\subset B(H) [[/math]]
  with the parameter being the index of the subfactor, [math]N=[A_1,A_0][/math].
• The collection [math]P=(P_k)[/math] formed by the linear spaces
  [[math]] P_k=A_0'\cap A_k [[/math]]
  which contains the image of [math]TL_N[/math], has a planar algebra structure.
• The index [math]N=[A_1,A_0][/math], which is by definition a Murray-von Neumann continuous quantity [math]N\in[1,\infty][/math], must satisfy the following condition:
  [[math]] N\in\left\{4\cos^2\left(\frac{\pi}{n}\right)\Big|n\in\mathbb N\right\}\cup[4,\infty] [[/math]]
  That is, in the small index range, the index of subfactors is quantized.

Show Proof

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

(1) The idea here is that the functional analytic study of the basic construction leads to the conclusion that the sequence of projections [math]e_1,e_2,e_3,\ldots\in B(H)[/math] behaves algebrically exactly as the rescaled sequence of diagrams [math]\varepsilon_1,\varepsilon_2,\varepsilon_3,\ldots\in TL_N[/math] given by:

[[math]] \varepsilon_1={\ }^\cup_\cap [[/math]]

[[math]] \varepsilon_2=|\!{\ }^\cup_\cap [[/math]]

[[math]] \varepsilon_3=||\!{\ }^\cup_\cap [[/math]]

[[math]] \vdots [[/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.

(2) Since the orthogonal 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 [math]e_1,\ldots,e_{k-1}\in P_k[/math] for any [math]k[/math], and so we have:

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

The point now is that the planar algebra structure of [math]TL_N[/math], obtained by composing diagrams, can be shown to extend into an abstract planar algebra structure of [math]P[/math].

(3) This is something quite surprising, which follows from (1), via some clever positivity considerations, involving the Perron-Frobenius theorem. In order to best comment on what happens, 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]] \vdots [[/math]]

In order to prove now the result, the first observation is that, 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 certain orthogonal polynomials. In practice now, the most elegant way of proving the result is by using the fundamental fact, from (1), 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 above. Let us mention too that, at a more advanced level, the subfactors having index [math]N\in[1,4][/math] can be classified by ADE diagrams, and the obstruction [math]N=4\cos^2(\frac{\pi}{n})[/math] itself comes from the fact that [math]N[/math] must be the squared norm of such a graph.

■

As before with other advanced operator algebra topics, our explanations here were quite brief. For more on all this, we recommend Jones' original paper , then his statistical mechanics paper too, and then his planar algebra paper .

General references

Banica, Teo (2024). "Invitation to Hadamard matrices". arXiv:1910.06911 [math.CO].