11d. Abelian subalgebras

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

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We would like to end this chapter with something more refreshing, in relation with the above, namely matrix models, and abelian subalgebras. These two topics are actually related, and in a quite subtle way, and we will provide an introduction to this.

Let us first discuss the matrix models, for the quantum groups. One interesting method for the study of the closed subgroups [math]G\subset U_N^+[/math] consists in modelling the coordinates [math]u_{ij}\in C(G)[/math] by concrete variables [math]U_{ij}\in B[/math]. Indeed, assuming that the model is faithful in some suitable sense, and that the target algebra [math]B[/math] is something quite familiar, all questions about [math]G[/math] would correspond in this way to routine questions inside [math]B[/math].

Regarding now the choice of the target algebra [math]B[/math], some very familiar and convenient algebras are the random matrix ones, [math]B=M_K(C(T))[/math], with [math]K\in\mathbb N[/math], and [math]T[/math] being a compact space. We are led in this way to the following definition:

Definition

A matrix model for [math]G\subset U_N^+[/math] is a morphism of [math]C^*[/math]-algebras

[[math]] \pi:C(G)\to M_K(C(T)) [[/math]]

where [math]T[/math] is a compact space, and [math]K\geq1[/math] is an integer.

There are many examples of such models, and will discuss them later on. For the moment, let us develop some general theory. The question to be solved is that of understanding the suitable faithfulness assumptions needed on [math]\pi[/math], as for the model to “remind” the quantum group. The simplest situation is when [math]\pi[/math] is faithful in the usual sense. Let us introduce the following notion, which is related to faithfulness:

Definition

A matrix model [math]\pi:C(G)\to M_K(C(T))[/math] is called stationary when

[[math]] \int_G=\left(tr\otimes\int_T\right)\pi [[/math]]

where [math]\int_T[/math] is the integration with respect to a given probability measure on [math]T[/math].

Here the term “stationary” comes from a functional analytic interpretation of all this, with a certain Cesàro limit being needed to be stationary, and this will be explained later. Yet another explanation comes from a certain relation with the lattice models, but this relation is rather something folklore, not axiomatized yet. We will be back to this.

As a first result now, which is something which is not exactly trivial, and whose proof requires some functional analysis, the stationarity property implies the faithfulness:

Theorem

Assuming that [math]G\subset U_N^+[/math] has a stationary model,

[[math]] \pi:C(G)\to M_K(C(T))\quad,\quad \int_G=\left(tr\otimes\int_T\right)\pi [[/math]]

it follows that [math]G[/math] is coamenable, and that the model is faithful, coming as:

[[math]] C(G)\subset L^\infty(G)\subset M_K(L^\infty(T)) [[/math]]

Moreover, we can have such models only when the algebra [math]L^\infty(G)[/math] is of type [math]{\rm I}[/math].

Show Proof

We use the basic theory of compact and discrete quantum groups, developed in chapter 7. Assume that we have a stationary model, as in the statement. By performing the GNS construction with respect to [math]\int_G[/math], we obtain a factorization as follows, which commutes with the respective canonical integration functionals:

[[math]] \pi:C(G)\to C(G)_{red}\subset M_K(C(T)) [[/math]]

Thus, in what regards the coamenability question, we can assume that [math]\pi[/math] is faithful. With this assumption made, observe that we have embeddings as follows:

[[math]] C^\infty(G)\subset C(G)\subset M_K(C(T)) [[/math]]

The point now is that the GNS construction gives a better embedding, as follows:

[[math]] L^\infty(G)\subset M_K(L^\infty(T)) [[/math]]

Now since the von Neumann algebra on the right is of type I, so must be its subalgebra [math]A=L^\infty(G)[/math]. This means that, when writing the center of this latter algebra as [math]Z(A)=L^\infty(X)[/math], the whole algebra decomposes over [math]X[/math], as an integral of type I factors:

[[math]] L^\infty(G)=\int_XM_{K_x}(\mathbb C)\,dx [[/math]]

In particular, we can see from this that [math]C^\infty(G)\subset L^\infty(G)[/math] has a unique [math]C^*[/math]-norm, and so [math]G[/math] is coamenable. Finally, the other assertions follow as well from the above, because our factorization of [math]\pi[/math] consists of the identity, and of an inclusion.

■

More generally now, we can talk about matrix models for the algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math], in the obvious way, and we have the following result:

Theorem

Given a matrix model [math]\pi:C(X)\to M_K(C(T))[/math], with both [math]X,T[/math] being assumed to have integration functionals, the following are equivalent:

• [math]\pi[/math] is stationary, in the sense that [math]\int_X=(tr\otimes\smallint_T)\pi[/math].
• [math]\pi[/math] produces an inclusion [math]\pi':C_{red}(X)\subset M_K(X(T))[/math].
• [math]\pi[/math] produces an inclusion [math]\pi'':L^\infty(X)\subset M_K(L^\infty(T))[/math].

Moreover, in the quantum group case, these conditions imply that [math]\pi[/math] is faithful.

Show Proof

Consider the following diagram, with all the solid arrows being by definition the canonical maps between the various algebras concerned:

[[math]] \xymatrix@R=60pt@C=30pt{ M_K(C(T))\ar[rrr]&&&M_K(L^\infty(T))\\ C(X)\ar[u]^\pi\ar[r]&C_{red}(X)\ar[rr]\ar@.[ul]_{\pi'}&&L^\infty(X)\ar@.[u]^{\pi''} } [[/math]]

With this picture in hand, the equivalences [math](1)\iff(2)\iff(3)[/math] between the above conditions (1,2,3) are all clear, coming from the basic properties of the GNS construction. As for the last assertion, this is something that we know from Theorem 11.27.

■

Moving ahead now, our claim is that our modelling philosophy, with type [math]{\rm I}[/math] algebras as target, and more specifically with random matrix algebras as target, can perfectly apply, at least in the quantum group case, to the type [math]{\rm II}[/math] algebras as well.

We have indeed the following result, which is something quite subtle:

Theorem

Given a matrix model [math]\pi:C(G)\to M_K(C(T))[/math], with [math]T[/math] being a probability space, there exists a smallest subgroup [math]G'\subset G[/math] producing a factorization

[[math]] \pi:C(G)\to C(G')\to M_K(C(T)) [[/math]]

with the intermediate algebra [math]C(G')[/math] being called Hopf image of [math]\pi[/math]. When [math]\pi[/math] is inner faithful, in the sense that we have [math]G=G'[/math], we have the formula

[[math]] \int_G=\lim_{k\to\infty}\sum_{r=1}^k\varphi^{*r} [[/math]]

where [math]\varphi=(tr\otimes\smallint_T)\pi[/math] is the matrix model trace, and where [math]\phi*\psi=(\phi\otimes\psi)\Delta[/math]. Also, the model [math]\pi[/math] is stationary precisely when this latter convergence is stationary.

Show Proof

All this is well-known, the idea being as follows:

(1) The construction of the Hopf image can be done by dividing the algebra [math]C(G)[/math] by a suitable ideal, but for our purposes here it is more convenient to go via an alternative proof. Let us denote by [math]u=(u_{ij})[/math] the fundamental corepresentation of [math]G[/math], and consider the following vector spaces, taken in a formal sense, where [math]U_{ij}=\pi(u_{ij})[/math]:

[[math]] C_{kl}=Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

Since the morphisms increase the intertwining spaces, when defined either in a representation theory sense, or just formally, we have inclusions as follows:

[[math]] Hom(u^{\otimes k},u^{\otimes l})\subset Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

More generally, we have such inclusions when replacing [math](G,u)[/math] with any pair producing a factorization of [math]\pi[/math]. Thus, by Woronowicz's Tannakian duality , the Hopf image must be given by the fact that the intertwining spaces must be the biggest, subject to the above inclusions. But since [math]u[/math] is biunitary, so is [math]U[/math], and it follows that the above spaces [math]C_{kl}[/math] form a Tannakian category, so have a quantum group [math](G',v)[/math] given by:

[[math]] Hom(v^{\otimes k},v^{\otimes l})=Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

By the above discussion, [math]C(G')[/math] follows to be the Hopf image of [math]\pi[/math], as claimed.

(2) The formula for [math]\int_G[/math] follows by adapting Woronowicz's construction of the Haar integration functional, from , to the matrix model situation. If we denote by [math]\int_G'[/math] the limit in the statement, we must prove that this limit converges, and that we have:

[[math]] \int_G'=\int_G [[/math]]

It is enough to check this on the coefficients of corepresentations, and if we let [math]w=u^{\otimes k}[/math] be one of the Peter-Weyl corepresentations, we must prove that we have:

[[math]] \left(id\otimes\int_G'\right)w=\left(id\otimes\int_G\right)w [[/math]]

We know from chapter 7 that the matrix on the right is the orthogonal projection onto [math]Fix(w)[/math]. Regarding now the matrix on the left, this is the orthogonal projection onto the [math]1[/math]-eigenspace of [math](id\otimes\varphi\pi)w[/math]. Now observe that, if we set [math]W_{ij}=\pi(w_{ij})[/math], we have:

[[math]] (id\otimes\varphi\pi)w=(id\otimes\varphi)W [[/math]]

Thus, exactly as in chapter 7, we conclude that the [math]1[/math]-eigenspace that we are interested in equals [math]Fix(W)[/math]. But, according to the proof of (1) above, we have:

[[math]] Fix(W)=Fix(w) [[/math]]

Thus, we have proved that we have [math]\int_G'=\int_G[/math], as desired.

■

The above result, with contributions by many people, and we refer to for the story, is quite important, for many reasons, mainly coming from the following fact: \begin{fact} There is no known restriction on the quantum groups having a model

[[math]] \pi:C(G)\to M_K(C(T)) [[/math]]

which is inner faithful, in the above sense. \end{fact} Which is obviously something interesting, conjecturally making Theorem 11.29 a clever way of passing from type [math]{\rm II}[/math] to type [math]{\rm I}[/math]. There are also connections here with the Connes embedding problem, and with all sorts of questions from algebra, geometry, analysis and probability, coming from both mathematics and physics. We will be back to this.

In the general quantum algebraic manifold setting now, talking about inner faithfulness is in general not possible, unless our manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] has some extra special structure, as for instance being an affine homogeneous space, and we refer here to .

Changing topics now, let us go back to the arbitrary von Neumann algebras, and explore some further perspectives opened by the various results that we know. Given a von Neumann algebra [math]A\subset B(H)[/math], looking at the center [math]Z(A)=A\cap A'[/math] is not the only possible way of getting to commutative, or abelian subalgebras, and we have as well:

Definition

Given a von Neumann algebra [math]A\subset B(H)[/math], an abelian subalgebra [math]B\subset A[/math] which is maximal, in the sense that there is no bigger abelian algebra

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

is called maximal abelian subalgebra (MASA).

It is possible to say many interesting things about the MASA, and skipping some details here, if we want to further build on this notion, we are led to:

Definition

Given a von Neumann algebra [math]A[/math] coming with a trace [math]tr:A\to\mathbb C[/math], assume that we have a pair of maximal abelian subalgebras

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

satisfying the following orthogonality condition, with respect to the trace:

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

We say then that [math]B,C[/math] are orthogonal maximal abelian subalgebras.

Here the scalar product is by definition [math] \lt b,c \gt =tr(bc^*)[/math], and by taking into account the multiples of the identity, the orthogonality condition reformulates as follows:

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

As explained by Popa in , the interest in Definition 11.32 comes from the fact that a pair of orthogonal MASA brings some sort of 2D orientation inside the von Neumann algebra [math]A[/math], or at least inside the subalgebra [math] \lt B,C \gt \subset A[/math] generated by the MASA. There is also an obvious link with the notion of noncommutative independence discussed in chapter 8. But more on all this later, in chapter 15 below, when doing subfactors.

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], endowed with its usual matrix trace. And in this context, we have the following surprising result of Popa :

Theorem

Up to a conjugation by a unitary, the pairs of orthogonal maximal abelian subalgebras in the simplest factor, namely [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, in the sense that [math]|H_{ij}|=1[/math] for any [math]i,j[/math], and the rows of [math]H[/math] are pairwise orthogonal.

Show Proof

Any maximal abelian subalgebra in [math]M_N(\mathbb C)[/math] being conjugated to [math]\Delta[/math], we can assume, up to conjugation by a unitary, that we have, with [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]]

Thus the rescaled matrix [math]H=\sqrt{N}U[/math] must satisfy the following condition:

[[math]] |H_{ij}|=1 [[/math]]

Thus, we are led to the conclusion in the statement.

■

The Hadamard matrices appearing in Theorem 11.33 are well-known objects, appearing in several branches of combinatorics, and quantum physics. The basic examples of such matrices are the Fourier matrices of abelian groups, constructed as follows:

Theorem

Given a finite abelian group [math]G[/math], with dual group [math]\widehat{G}=\{\chi:G\to\mathbb T\}[/math], consider the Fourier coupling [math]\mathcal F_G:G\times\widehat{G}\to\mathbb T[/math], given by [math](i,\chi)\to\chi(i)[/math].

• Via the standard isomorphism [math]G\simeq\widehat{G}[/math], this Fourier coupling can be regarded as a square matrix, [math]F_G\in M_G(\mathbb T)[/math], which is a complex Hadamard matrix.
• For the cyclic group [math]G=\mathbb Z_N[/math] we obtain in this way, via the standard identification [math]\mathbb Z_N=\{1,\ldots,N\}[/math], the standard Fourier matrix, [math]F_N=(w^{ij})[/math] with [math]w=e^{2\pi i/N}[/math].
• In general, when using a decomposition [math]G=\mathbb Z_{N_1}\times\ldots\times\mathbb Z_{N_k}[/math], the corresponding Fourier matrix is given by [math]F_G=F_{N_1}\otimes\ldots\otimes F_{N_k}[/math].

Show Proof

This follows indeed from some basic facts from group theory:

(1) With the identification [math]G\simeq\widehat{G}[/math] made our matrix is given by [math](F_G)_{i\chi}=\chi(i)[/math], and the scalar products between the rows are computed as follows:

[[math]] \lt R_i,R_j \gt =\sum_\chi\chi(i)\overline{\chi(j)} =\sum_\chi\chi(i-j) =|G|\cdot\delta_{ij} [[/math]]

Thus, we obtain indeed a complex Hadamard matrix.

(2) This follows from the well-known and elementary fact that, via the identifications [math]\mathbb Z_N=\widehat{\mathbb Z_N}=\{1,\ldots,N\}[/math], the Fourier coupling here is as follows, with [math]w=e^{2\pi i/N}[/math]:

[[math]] (i,j)\to w^{ij} [[/math]]

(3) We use here the following well-known formula, for the duals of products:

[[math]] \widehat{H\times K}=\widehat{H}\times\widehat{K} [[/math]]

At the level of the corresponding Fourier couplings, we obtain from this:

[[math]] F_{H\times K}=F_H\otimes F_K [[/math]]

Now by decomposing [math]G[/math] into cyclic groups, as in the statement, and by using (2) for the cyclic components, we obtain the formula in the statement.

■

Summarizing, we have some interesting connections with finite group theory, and with the associated Fourier matrices. However, there are as well many exotic examples of Hadamard matrices, nor necessarily coming from finite groups, as in Theorem 11.34, and all this is quite of interest for us, in connection with Theorem 11.33.

We will be back to this later, with more results on the subject, in chapters 13-16, when talking about subfactors. Among others, we will see there that the combinatorics of the MASA associated to an Hadamard matrix comes from a certain quantum permutation groups, appearing as in Theorem 11.29, via a matrix model. More on this soon.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].