13a. Operator algebras

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

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Welcome to this fourth and last part of the present book. We discuss here yet another idea in order to deal with the Hadamard matrices, be them real or complex, this time in relation with quantum groups. What we will be doing here will be deeply related to all sorts of advanced algebraic considerations regarding the Hadamard matrices, from chapters 1-12 above, and also to a quite good deal of deep considerations from operator algebras, following Haagerup , Jones , Popa and others. So, we will be here working at a foundational level in mathematical physics. In fact, all the potential applications of the complex Hadamard matrices to questions in physics, be them from general quantum mechanics, quantum information, statistical mechanics, and many more, are expected to come via the link with the quantum groups.

The idea is extremely simple, namely that associated to any complex Hadamard matrix [math]H\in M_N(\mathbb C)[/math] is a certain quantum permutation group [math]G\subset S_N^+[/math], which describes the “symmetries” of the matrix. As a basic illustration, for a Fourier matrix [math]H=F_G[/math] we obtain the group [math]G[/math] itself, acting on itself, [math]G\subset S_G[/math]. In general, however, we obtain non-classical quantum groups, whose computation is a key problem.

In order to discuss this, we will need many preliminaries, namely operator theory, operator algebras and quantum spaces, compact quantum groups, quantum permutation groups, and finally matrix models for such quantum groups, which produce the above correspondence. Before getting started, some references. For functional analysis, operator theory and operator algebras you have Lax , and also Connes , if you want to learn more. For quantum groups you have the papers of Woronowicz , or my book , but we will explain the needed material here. For tools for dealing with such quantum groups, these will often come from Jones , , and Voiculescu .

Also, importantly, there are no quantum groups or quantum mechanics without quantum mechanics. In order to appreciate what will follow, get to learn some, standard places being Feynman , Griffiths , Weinberg . In case you would rather enjoy a rigorous text written by a mathematician, you can go with my book . Although that is not an inch more clever, or even rigorous, than what physicists are doing.

Getting started now, we first have the following standard result:

Theorem

Given a complex Hilbert space [math]H[/math], the linear operators [math]T:H\to H[/math] which are bounded, in the sense that the quantity

[[math]] ||T||=\sup_{||x||\leq1}||Tx|| [[/math]]

is finite, form a complex algebra with unit, denoted [math]B(H)[/math], having the following properties:

• [math]B(H)[/math] is complete with respect to [math]||.||[/math], so we have a Banach algebra.
• [math]B(H)[/math] has an involution [math]T\to T^*[/math], given by [math] \lt Tx,y \gt = \lt x,T^*y \gt [/math].

In addition, the norm and involution are related by the formula [math]||TT^*||=||T||^2[/math].

Show Proof

The fact that we have indeed an algebra follows from:

[[math]] ||S+T||\leq||S||+||T||\quad,\quad ||\lambda T||=|\lambda|\cdot||T||\quad,\quad ||ST||\leq||S||\cdot||T|| [[/math]]

Regarding now (1), if [math]\{T_n\}\subset B(H)[/math] is Cauchy then [math]\{T_nx\}[/math] is Cauchy for any [math]x\in H[/math], so we can define the limit [math]T=\lim_{n\to\infty}T_n[/math] simply by setting:

[[math]] Tx=\lim_{n\to\infty}T_nx [[/math]]

As for (2), here the existence of [math]T^*[/math] comes from the fact that [math]\varphi(x)= \lt Tx,y \gt [/math] being a linear map [math]H\to\mathbb C[/math], we must have, for a certain vector [math]T^*y\in H[/math]:

[[math]] \varphi(x)= \lt x,T^*y \gt [[/math]]

Moreover, since this vector is unique, [math]T^*[/math] is unique too, and we have as well:

[[math]] (S+T)^*=S^*+T^*\quad,\quad (\lambda T)^*=\bar{\lambda}T^*\quad,\quad (ST)^*=T^*S^*\quad,\quad (T^*)^*=T [[/math]]

Observe also that we have indeed [math]T^*\in B(H)[/math], because:

[[math]] \begin{eqnarray*} ||T|| &=&\sup_{||x||=1}\sup_{||y||=1} \lt Tx,y \gt \\ &=&\sup_{||y||=1}\sup_{||x||=1} \lt x,T^*y \gt \\ &=&||T^*|| \end{eqnarray*} [[/math]]

Regarding now the last assertion, observe first that we have:

[[math]] ||TT^*|| \leq||T||\cdot||T^*|| =||T||^2 [[/math]]

On the other hand, we have as well the following estimate:

[[math]] \begin{eqnarray*} ||T||^2 &=&\sup_{||x||=1}| \lt Tx,Tx \gt |\\ &=&\sup_{||x||=1}| \lt x,T^*Tx \gt |\\ &\leq&||T^*T|| \end{eqnarray*} [[/math]]

By replacing [math]T\to T^*[/math] we obtain from this [math]||T||^2\leq||TT^*||[/math], and we are done.

■

We will be interested in the algebras of operators, rather than in the operators themselves. The basic axioms here, inspired from Theorem 13.1, are as follows:

Definition

A [math]C^*[/math]-algebra is a complex algebra with unit [math]A[/math], having:

• A norm [math]a\to||a||[/math], making it a Banach algebra (the Cauchy sequences converge).
• An involution [math]a\to a^*[/math], which satisfies [math]||aa^*||=||a||^2[/math], for any [math]a\in A[/math].

According to Theorem 13.1, the operator algebra [math]B(H)[/math] itself is a [math]C^*[/math]-algebra. More generally, we have as examples all the closed [math]*[/math]-subalgebras [math]A\subset B(H)[/math]. We will see later on (the “GNS theorem”) that any [math]C^*[/math]-algebra appears in fact in this way. However, even before knowing that, in view of the examples that we have, we can think of the elements [math]a\in A[/math] of an arbitrary [math]C^*[/math]-algebra as being some kind of “generalized beounded operators”, on some Hilbert space which is not necessarily present. By using this idea, one can emulate spectral theory in this setting, and we have the following result:

Theorem

Given [math]a\in A[/math], define its spectrum as being the set

[[math]] \sigma(a)=\left\{\lambda\in\mathbb C\Big|a-\lambda\not\in A^{-1}\right\} [[/math]]

and its spectral radius [math]\rho(a)[/math] as the radius of the smallest centered disk containing [math]\sigma(a)[/math].

• The spectrum of a norm one element is in the unit disk.
• The spectrum of a unitary element [math](a^*=a^{-1}[/math]) is on the unit circle.
• The spectrum of a self-adjoint element ([math]a=a^*[/math]) consists of real numbers.
• The spectral radius of a normal element ([math]aa^*=a^*a[/math]) is equal to its norm.

Show Proof

Our first claim is that for any polynomial [math]f\in\mathbb C[X][/math], and more generally for any rational function [math]f\in\mathbb C(X)[/math] having poles outside [math]\sigma(a)[/math], we have:

[[math]] \sigma(f(a))=f(\sigma(a)) [[/math]]

This indeed something well-known for the usual matrices. In the general case, assume first that we have a polynomial, [math]f\in\mathbb C[X][/math]. If we pick an arbitrary number [math]\lambda\in\mathbb C[/math], and write [math]f(X)-\lambda=c(X-r_1)\ldots(X-r_k)[/math], we have then, as desired:

[[math]] \begin{eqnarray*} \lambda\notin\sigma(f(a)) &\iff&f(a)-\lambda\in A^{-1}\\ &\iff&c(a-r_1)\ldots(a-r_k)\in A^{-1}\\ &\iff&a-r_1,\ldots,a-r_k\in A^{-1}\\ &\iff&r_1,\ldots,r_k\notin\sigma(a)\\ &\iff&\lambda\notin f(\sigma(a)) \end{eqnarray*} [[/math]]

Assume now that we are in the general case, [math]f\in\mathbb C(X)[/math]. We pick [math]\lambda\in\mathbb C[/math], we write [math]f=P/Q[/math], and we consider the following polynomial:

[[math]] F=P-\lambda Q [[/math]]

By using the above finding, for this polynomial [math]F[/math], we obtain, as desired:

[[math]] \begin{eqnarray*} \lambda\in\sigma(f(a)) &\iff&F(a)\notin A^{-1}\\ &\iff&0\in\sigma(F(a))\\ &\iff&0\in F(\sigma(a))\\ &\iff&\exists\mu\in\sigma(a),F(\mu)=0\\ &\iff&\lambda\in f(\sigma(a)) \end{eqnarray*} [[/math]]

Regarding now the assertions in the statement, these basically follow from this:

(1) This comes from the following formula, valid when [math]||a|| \lt 1[/math]:

[[math]] \frac{1}{1-a}=1+a+a^2+\ldots [[/math]]

(2) Assuming [math]a^*=a^{-1}[/math], we have the following norm computations:

[[math]] ||a||=\sqrt{||aa^*||}=\sqrt{1}=1 [[/math]]

[[math]] ||a^{-1}||=||a^*||=||a||=1 [[/math]]

If we denote by [math]D[/math] the unit disk, we obtain from this, by using (1):

[[math]] ||a||=1\implies\sigma(a)\subset D [[/math]]

[[math]] ||a^{-1}||=1\implies\sigma(a^{-1})\subset D [[/math]]

On the other hand, by using the rational function [math]f(z)=z^{-1}[/math], we have:

[[math]] \sigma(a^{-1})\subset D\implies \sigma(a)\subset D^{-1} [[/math]]

Now by putting everything together we obtain, as desired:

[[math]] \sigma(a)\subset D\cap D^{-1}=\mathbb T [[/math]]

(3) This follows by using (2), and the following rational function, with [math]t\in\mathbb R[/math]:

[[math]] f(z)=\frac{z+it}{z-it} [[/math]]

Indeed, for [math]t \gt \gt 0[/math] the element [math]f(a)[/math] is well-defined, and we have:

[[math]] \left(\frac{a+it}{a-it}\right)^* =\frac{a-it}{a+it} =\left(\frac{a+it}{a-it}\right)^{-1} [[/math]]

Thus [math]f(a)[/math] is a unitary, and by (2) its spectrum is contained in [math]\mathbb T[/math]. We conclude that we have [math]f(\sigma(a))=\sigma(f(a))\subset\mathbb T[/math], and so [math]\sigma(a)\subset f^{-1}(\mathbb T)=\mathbb R[/math], as desired.

(4) We have [math]\rho(a)\leq ||a||[/math] from (1). Conversely, given [math]\rho \gt \rho(a)[/math], we have:

[[math]] \int_{|z|=\rho}\frac{z^n}{z-a}\,dz =\sum_{k=0}^\infty\left(\int_{|z|=\rho}z^{n-k-1}dz\right) a^k =a^{n-1} [[/math]]

By applying the norm and taking [math]n[/math]-th roots we obtain:

[[math]] \rho\geq\lim_{n\to\infty}||a^n||^{1/n} [[/math]]

In the case [math]a=a^*[/math] we have [math]||{a^n}||=||{a}||^n[/math] for any exponent of the form [math]n=2^k[/math], and by taking [math]n[/math]-th roots we get [math]\rho\geq ||{a}||[/math]. This gives the missing inequality, namely:

[[math]] \rho(a)\geq ||a|| [[/math]]

In the general case [math]aa^*=a^*a[/math] we have [math]a^n(a^n)^*=(aa^*)^n[/math], and we get:

[[math]] \rho(a)^2=\rho(aa^*) [[/math]]

Now since [math]aa^*[/math] is self-adjoint, we get [math]\rho(aa^*)=||{a}||^2[/math], and we are done.

■

With these preliminaries in hand, we can now formulate some theorems. The basic facts about the [math]C^*[/math]-algebras, that we will need here, can be summarized as:

Theorem

The [math]C^*[/math]-algebras have the following properties:

• The commutative ones are those of the form [math]C(X)[/math], with [math]X[/math] compact space.
• Any such algebra [math]A[/math] embeds as [math]A\subset B(H)[/math], for some Hilbert space [math]H[/math].
• In finite dimensions, these are the direct sums of matrix algebras.

Show Proof

All this is standard, the idea being as follows:

(1) Given a compact space [math]X[/math], the algebra [math]C(X)[/math] of continuous functions [math]f:X\to\mathbb C[/math] is indeed a [math]C^*[/math]-algebra, with norm and involution as follows:

[[math]] ||f||=\sup_{x\in X}|f(x)|\quad,\quad f^*(x)=\overline{f(x)} [[/math]]

Observe that this algebra is indeed commutative, because:

[[math]] f(x)g(x)=g(x)f(x) [[/math]]

Conversely, if [math]A[/math] is commutative, we can define [math]X=Spec(A)[/math] to be the space of all characters [math]\chi :A\to\mathbb C[/math], with the topology making continuous all the evaluation maps [math]ev_a:\chi\to\chi(a)[/math]. We have then a morphism of algebras, as follows:

[[math]] ev:A\to C(X)\quad,\quad a\to ev_a [[/math]]

Theorem 13.3 (3) shows that [math]ev[/math] is a [math]*[/math]-morphism, Theorem 13.3 (4) shows that [math]ev[/math] is isometric, and finally the Stone-Weierstrass theorem shows that [math]ev[/math] is surjective.

(2) This is standard for [math]A=C(X)[/math], where we can pick a probability measure on [math]X[/math], and set [math]H=L^2(X)[/math], and use the following embedding:

[[math]] A\subset B(H)\quad,\quad f\to(g\to fg) [[/math]]

In the general case, where [math]A[/math] is no longer commutative, the proof is quite similar, by emulating basic measure theory in the abstract [math]C^*[/math]-algebra setting.

(3) Assuming that [math]A[/math] is finite dimensional, we can first decompose its unit as follows, with [math]p_i\in A[/math] being central minimal projections:

[[math]] 1=p_1+\ldots+p_k [[/math]]

Each of the linear spaces [math]A_i=p_iAp_i[/math] is then a non-unital [math]*[/math]-subalgebra of [math]A[/math], and we have a non-unital [math]*[/math]-algebra sum decomposition, as follows:

[[math]] A=A_1\oplus\ldots\oplus A_k [[/math]]

On the other hand, since each central projection [math]p_i[/math] was assumed minimal, we have unital [math]*[/math]-algebra isomorphisms as follows, with [math]r_i=rank(p_i)[/math]:

[[math]] A_i\simeq M_{r_i}(\mathbb C) [[/math]]

Thus, we obtain an isomorphism [math]A\simeq M_{r_1}(\mathbb C)\oplus\ldots\oplus M_{r_k}(\mathbb C)[/math], as desired.

■

All the above was of course quite brief, but full details on this can be found in any book on functional analysis, as for instance Lax . In what concerns us, we will be mainly interested in Theorem 13.4 (1), called Gelfand theorem, which suggests formulating:

Definition

Given a [math]C^*[/math]-algebra [math]A[/math], not necessarily commutative, we write

[[math]] A=C(X) [[/math]]

and call the abstract object [math]X[/math] a compact quantum space.

In other words, we define the category of the compact quantum spaces [math]X[/math] to be the category of the [math]C^*[/math]-algebras [math]A[/math], with the arrows reversed. Due to the Gelfand theorem, 13.4 (1) above, the category of the usual compact spaces embeds covariantly into the category of the compact quantum spaces, and the image of this embedding consists precisely of the compact quantum spaces [math]X[/math] which are “classical”, in the sense that the corresponding [math]C^*[/math]-algebra [math]A=C(X)[/math] is commutative. Thus, what we have done here is to extend the category of the usual compact spaces, and this justifies Definition 13.5.

In practice now, the general compact quantum spaces [math]X[/math] do not have points, but we can perfectly study them via the associated algebras [math]A=C(X)[/math], a bit in the same way as we study a compact Lie group via its associated Lie algebra, or an algebraic manifold via the ideal of polynomials vanishing on it, and so on. In short, nothing that much abstract going on here, just another instance of the old idea “we will use algebras, no need for points”, with the remark that for us, the use of points will be actually forbidden.

General references

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