7a. Gelfand theorem

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We have seen that the von Neumann algebras [math]A\subset B(H)[/math] are interesting objects, and it is tempting to go ahead with a systematic study of such algebras. This is what Murray and von Neumann did, when first coming across such algebras, back in the 1930s, in their series of papers , , , , , . In what concerns us, we will rather keep this material for later, and talk instead, in this chapter and in the next one, of things which are perhaps more basic, motivated by the following definition:

Definition

Given a von Neumann algebra [math]A\subset B(H)[/math], coming with a faithful positive unital trace [math]tr:A\to\mathbb C[/math], we write

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

and call [math]X[/math] a quantum probability space. We also write the trace as [math]tr=\int_X[/math], and call it integration with respect to the uniform measure on [math]X[/math].

Obviously, this is something exciting, and we have seen how some interesting theory can be developed along these lines in the simplest case, that of the random matrix algebras. Thus, all this needs a better understanding, before going ahead with the above-mentioned Murray-von Neumann theory. In order to get started, here are a few comments:

(1) Generally speaking, all this comes from the fact that the commutative von Neumann algebras are those of the form [math]A=L^\infty(X)[/math], with [math]X[/math] being a measured space. Since in the finite measure case, [math]\mu(X) \lt \infty[/math], the integration can be regarded as being a faithful positive unital trace [math]tr:L^\infty(X)\to\mathbb C[/math], we are basically led to Definition 7.1.

(2) Regarding our assumption [math]\mu(X) \lt \infty[/math], making the integration [math]tr:A\to\mathbb C[/math] bounded, this is something advanced, coming from deep classification results of von Neumann and Connes, which roughly state that “modulo classical measure theory, the study of the quantum measured spaces [math]X[/math] basically reduces to the case [math]\mu(X) \lt \infty[/math]”.

(3) Finally, the traciality of [math]tr:A\to\mathbb C[/math] is something advanced too, again coming from that classification results of von Neumann and Connes, which in their more precise formulation state that “modulo classical measure theory, the study of the quantum measured spaces [math]X[/math] basically reduces to the case where [math]\mu(X) \lt \infty[/math], and [math]\int_X[/math] is a trace”.

In short, complicated all this, and you will have to trust me here. Moving ahead now, there is one more thing to be discussed in connection with Definition 7.1, and this is physics. Let me formulate here the question that you surely have in mind: \begin{question} As physicists we already agreed, without clear evidence, that our operators [math]T:H\to H[/math] should be bounded. But what about quantum spaces, is it a good idea to assume that these are as above, of finite mass, and with tracial integration? \end{question} Well, this is certainly an interesting question. In favor of my choice, I would argue that the mathematical physics of Jones , , , , and Voiculescu , , needs a trace [math]tr:A\to\mathbb C[/math], as above. And the same goes for certain theoretical physics continuations of the main work of Connes , as for instance the basic theory of the Standard Model spectral triple of Chamseddine-Connes, whose free gauge group has tracial Haar integration. Needless to say, all this is quite subjective. But hey, question of theoretical physics you asked, answer of theoretical physics is what you get.

Hang on, we are not done yet. Now that we are convinced that Definition 7.1 is the correct one, be that on mathematical or physical grounds, let us look for examples. And here the situation is quite grim, because even in the classical case, we have: \begin{fact} The measure on a classical measured space [math]X[/math] cannot come out of nowhere, and is usually a Haar measure, appearing by theorem. Thus, in our picture

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

both the Hilbert space [math]H=L^2(X)[/math] and the von Neumann algebra [math]A=L^\infty(X)[/math] should appear by theorem, not by definition, contrary to what Definition 7.1 says. \end{fact} To be more precise, in what regards the first assertion, this is certainly the case with simple objects like Lie groups, or spheres and other homogeneous spaces. Of course you might say that [math][0,1][/math] with the uniform measure is a measured space, but isn't [math][0,1][/math] obtained by cutting the Lie group [math]\mathbb R[/math], with its Haar measure. And the same goes with [math][0,1][/math] with an arbitrary measure [math]f(x)dx[/math], or with [math][0,1][/math] being deformed into a curve, and so on, because that [math]dx[/math], or what is left from it, will always refer to the Haar measure of [math]\mathbb R[/math].

As for the second assertion, nothing much to comment here, mathematics has spoken. So, getting back now to Definition 7.1 as it is, looks like we have two dead bodies there, the Hilbert space [math]H[/math] and the operator algebra [math]A[/math]. So let us try to get rid of at least one of them. But which? In the lack of any obvious idea, let us turn to physics: \begin{question} In quantum mechanics, which came first, the Hilbert space [math]H[/math], or the operator algebra [math]A[/math]? \end{question} Unfortunately this question is as difficult as the one regarding the chicken and the egg. A look at what various physicists said on this matter, in a direct or indirect way, does not help much, and by the end of the day we are left with guidelines like “no one understands quantum mechanics” (Feynman), “shut up and compute” (Dirac) and so on. And all this, coming on top on what has been already said on Definition 7.1, of rather unclear nature, is probably too much. That is, the last drop, time to conclude: \begin{conclusion} The theory of von Neumann algebras has the same peculiarity as quantum mechanics: it tends to self-destruct, when approached axiomatically. \end{conclusion} And we will take this as good news, providing us with warm evidence that the theory of von Neumann algebras is indeed related to quantum mechanics. This is what matters, being on the right track, and difficulties and all the rest, we won't be scared by them.

Back to business now, in practice, we must go back to chapter 5, and examine what we were saying right before introducing the von Neumann algebras. And at that time, we were talking about general operator algebras [math]A\subset B(H)[/math], closed with respect to the norm, but not necessarily with respect to the weak topology. But this suggests formulating the following definition, somewhat as a purely mathematical answer to Question 7.4:

Definition

A [math]C^*[/math]-algebra is an complex algebra [math]A[/math], given with:

• A norm [math]a\to||a||[/math], making it into a Banach algebra.
• An involution [math]a\to a^*[/math], related to the norm by the formula [math]||aa^*||=||a||^2[/math].

Here by Banach algebra we mean a complex algebra with a norm satisfying all the conditions for a vector space norm, along with [math]||ab||\leq||a||\cdot||b||[/math] and [math]||1||=1[/math], and which is such that our algebra is complete, in the sense that the Cauchy sequences converge. As for the involution, this must be antilinear, antimultiplicative, and satisfying [math]a^{**}=a[/math].

As basic examples, we have the operator algebra [math]B(H)[/math], for any Hilbert space [math]H[/math], and more generally, the norm closed [math]*[/math]-subalgebras [math]A\subset B(H)[/math]. It is possible to prove that any [math]C^*[/math]-algebra appears in this way, but this is a non-trivial result, called GNS theorem, and more on this later. Note in passing that this result tells us that there is no need to memorize the above axioms for the [math]C^*[/math]-algebras, because these are simply the obvious things that can be said about [math]B(H)[/math], and its norm closed [math]*[/math]-subalgebras [math]A\subset B(H)[/math].

As a second class of basic examples, which are of great interest for us, we have:

Proposition

If [math]X[/math] is a compact space, the algebra [math]C(X)[/math] of continuous functions [math]f:X\to\mathbb C[/math] is a [math]C^*[/math]-algebra, with the usual norm and involution, namely:

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

This algebra is commutative, in the sense that [math]fg=gf[/math], for any [math]f,g\in C(X)[/math].

Show Proof

All this is clear from definitions. Observe that we have indeed:

[[math]] ||ff^*|| =\sup_{x\in X}|f(x)|^2 =||f||^2 [[/math]]

Thus, the axioms are satisfied, and finally [math]fg=gf[/math] is clear.

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In general, the [math]C^*[/math]-algebras can be thought of as being algebras of operators, over some Hilbert space which is not present. By using this philosophy, one can emulate spectral theory in this setting, with extensions of the various results from chapters 3,5:

Theorem

Given element [math]a\in A[/math] of a [math]C^*[/math]-algebra, define its spectrum as:

[[math]] \sigma(a)=\left\{\lambda\in\mathbb C\Big|a-\lambda\notin A^{-1}\right\} [[/math]]

The following spectral theory results hold, exactly as in the [math]A=B(H)[/math] case:

• We have [math]\sigma(ab)\cup\{0\}=\sigma(ba)\cup\{0\}[/math].
• We have polynomial, rational and holomorphic calculus.
• As a consequence, the spectra are compact and non-empty.
• The spectra of unitaries [math](u^*=u^{-1})[/math] and self-adjoints [math](a=a^*)[/math] are on [math]\mathbb T,\mathbb R[/math].
• The spectral radius of normal elements [math](aa^*=a^*a)[/math] is given by [math]\rho(a)=||a||[/math].

In addition, assuming [math]a\in A\subset B[/math], the spectra of [math]a[/math] with respect to [math]A[/math] and to [math]B[/math] coincide.

Show Proof

This is something that we know from chapter 3, in the case [math]A=B(H)[/math], and then from chapter 5, in the case [math]A\subset B(H)[/math]. In general, the proof is similar:

(1) Regarding the assertions (1-5), which are of course formulated a bit informally, the proofs here are perfectly similar to those for the full operator algebra [math]A=B(H)[/math]. All this is standard material, and in fact, things in chapters 3 were written in such a way as for their extension now, to the general [math]C^*[/math]-algebra setting, to be obvious.

(2) Regarding the last assertion, we know this from chapter 5 for [math]A\subset B\subset B(H)[/math], and the proof in general is similar. Indeed, the inclusion [math]\sigma_B(a)\subset\sigma_A(a)[/math] is clear. For the converse, assume [math]a-\lambda\in B^{-1}[/math], and consider the following self-adjoint element:

[[math]] b=(a-\lambda )^*(a-\lambda ) [[/math]]

The difference between the two spectra of [math]b\in A\subset B[/math] is then given by:

[[math]] \sigma_A(b)-\sigma_B(b)=\left\{\mu\in\mathbb C-\sigma_B(b)\Big|(b-\mu)^{-1}\in B-A\right\} [[/math]]

Thus this difference in an open subset of [math]\mathbb C[/math]. On the other hand [math]b[/math] being self-adjoint, its two spectra are both real, and so is their difference. Thus the two spectra of [math]b[/math] are equal, and in particular [math]b[/math] is invertible in [math]A[/math], and so [math]a-\lambda\in A^{-1}[/math], as desired.

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We can now get back to the commutative [math]C^*[/math]-algebras, and we have the following result, due to Gelfand, which will be of crucial importance for us:

Theorem

The commutative [math]C^*[/math]-algebras are exactly the algebras of the form

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

with the “spectrum” [math]X[/math] of such an algebra being the space of characters [math]\chi:A\to\mathbb C[/math], with topology making continuous the evaluation maps [math]ev_a:\chi\to\chi(a)[/math].

Show Proof

This is something that we basically know from chapter 5, but always good to talk about it again. Given a commutative [math]C^*[/math]-algebra [math]A[/math], we can define [math]X[/math] as in the statement. Then [math]X[/math] is compact, and [math]a\to ev_a[/math] is a morphism of algebras, as follows:

[[math]] ev:A\to C(X) [[/math]]

(1) We first prove that [math]ev[/math] is involutive. We use the following formula, which is similar to the [math]z=Re(z)+iIm(z)[/math] formula for the usual complex numbers:

[[math]] a=\frac{a+a^*}{2}+i\cdot\frac{a-a^*}{2i} [[/math]]

Thus it is enough to prove the equality [math]ev_{a^*}=ev_a^*[/math] for self-adjoint elements [math]a[/math]. But this is the same as proving that [math]a=a^*[/math] implies that [math]ev_a[/math] is a real function, which is in turn true, because [math]ev_a(\chi)=\chi(a)[/math] is an element of [math]\sigma(a)[/math], contained in [math]\mathbb R[/math].

(2) Since [math]A[/math] is commutative, each element is normal, so [math]ev[/math] is isometric:

[[math]] ||ev_a|| =\rho(a) =||a|| [[/math]]

(3) It remains to prove that [math]ev[/math] is surjective. But this follows from the Stone-Weierstrass theorem, because [math]ev(A)[/math] is a closed subalgebra of [math]C(X)[/math], which separates the points.

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In view of the Gelfand theorem, we can formulate the following key definition:

Definition

Given an arbitrary [math]C^*[/math]-algebra [math]A[/math], we write

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

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

This might look like something informal, but it is not. Indeed, we can define the category of compact quantum spaces to be the category of the [math]C^*[/math]-algebras, with the arrows reversed. When [math]A[/math] is commutative, the above space [math]X[/math] exists indeed, as a Gelfand spectrum, [math]X=Spec(A)[/math]. In general, [math]X[/math] is something rather abstract, and our philosophy here will be that of studying of course [math]A[/math], but formulating our results in terms of [math]X[/math]. For instance whenever we have a morphism [math]\Phi:A\to B[/math], we will write [math]A=C(X),B=C(Y)[/math], and rather speak of the corresponding morphism [math]\phi:Y\to X[/math]. And so on.

Technically speaking, we will see later that the above formalism has its limitations, and needs a fix. To be more precise, when looking at compact quantum spaces having a probability measure, there are more of them in the sense of Definition 7.10, than in the von Neumann algebra sense. Thus, all this needs a fix. But more on this later.

As a first concrete consequence of the Gelfand theorem, we have:

Proposition

Assume that [math]a\in A[/math] is normal, and let [math]f\in C(\sigma(a))[/math].

• We can define [math]f(a)\in A[/math], with [math]f\to f(a)[/math] being a morphism of [math]C^*[/math]-algebras.
• We have the “continuous functional calculus” formula [math]\sigma(f(a))=f(\sigma(a))[/math].

Show Proof

Since [math]a[/math] is normal, the [math]C^*[/math]-algebra [math] \lt a \gt [/math] that is generates is commutative, so if we denote by [math]X[/math] the space formed by the characters [math]\chi: \lt a \gt \to\mathbb C[/math], we have:

[[math]] \lt a \gt =C(X) [[/math]]

Now since the map [math]X\to\sigma(a)[/math] given by evaluation at [math]a[/math] is bijective, we obtain:

[[math]] \lt a \gt =C(\sigma(a)) [[/math]]

Thus, we are dealing with usual functions, and this gives all the assertions.

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As another consequence of the Gelfand theorem, we have:

Proposition

For a normal element [math]a\in A[/math], the following are equivalent:

• [math]a[/math] is positive, in the sense that [math]\sigma(a)\subset[0,\infty)[/math].
• [math]a=b^2[/math], for some [math]b\in A[/math] satisfying [math]b=b^*[/math].
• [math]a=cc^*[/math], for some [math]c\in A[/math].

Show Proof

This is very standard, exactly as in [math]A=B(H)[/math] case, as follows:

[math](1)\implies(2)[/math] Since [math]f(z)=\sqrt{z}[/math] is well-defined on [math]\sigma(a)\subset[0,\infty)[/math], we can set [math]b=\sqrt{a}[/math].

[math](2)\implies(3)[/math] This is trivial, because we can set [math]c=b[/math].

[math](3)\implies(1)[/math] We proceed by contradiction. By multiplying [math]c[/math] by a suitable element of [math] \lt cc^* \gt [/math], we are led to the existence of an element [math]d\neq0[/math] satisfying [math]-dd^*\geq0[/math]. By writing now [math]d=x+iy[/math] with [math]x=x^*,y=y^*[/math] we have:

[[math]] dd^*+d^*d=2(x^2+y^2)\geq0 [[/math]]

Thus [math]d^*d\geq0[/math], contradicting the fact that [math]\sigma(dd^*),\sigma(d^*d)[/math] must coincide outside [math]\{0\}[/math].

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Let us clarify now the relation between [math]C^*[/math]-algebras and von Neumann algebras. In order to do so, we need a prove a key result, called GNS representation theorem, stating that any [math]C^*[/math]-algebra appears as an operator algebra. As a first result, we have:

Proposition

Let [math]A[/math] be a commutative [math]C^*[/math]-algebra, write [math]A=C(X)[/math], with [math]X[/math] being a compact space, and let [math]\mu[/math] be a positive measure on [math]X[/math]. We have then

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

where [math]H=L^2(X)[/math], with [math]f\in A[/math] corresponding to the operator [math]g\to fg[/math].

Show Proof

Given a continuous function [math]f\in C(X)[/math], consider the operator [math]T_f(g)=fg[/math], on [math]H=L^2(X)[/math]. Observe that [math]T_f[/math] is indeed well-defined, and bounded as well, because:

[[math]] ||fg||_2 =\sqrt{\int_X|f(x)|^2|g(x)|^2d\mu(x)} \leq||f||_\infty||g||_2 [[/math]]

The application [math]f\to T_f[/math] being linear, involutive, continuous, and injective as well, we obtain in this way a [math]C^*[/math]-algebra embedding [math]A\subset B(H)[/math], as claimed.

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In order to prove the GNS representation theorem, we must extend the above construction, to the case where [math]A[/math] is not necessarily commutative. Let us start with:

Definition

Consider a [math]C^*[/math]-algebra [math]A[/math].

• [math]\varphi:A\to\mathbb C[/math] is called positive when [math]a\geq0\implies\varphi(a)\geq0[/math].
• [math]\varphi:A\to\mathbb C[/math] is called faithful and positive when [math]a\geq0,a\neq0\implies\varphi(a) \gt 0[/math].

In the commutative case, [math]A=C(X)[/math], the positive elements are the positive functions, [math]f:X\to[0,\infty)[/math]. As for the positive linear forms [math]\varphi:A\to\mathbb C[/math], these appear as follows, with [math]\mu[/math] being positive, and strictly positive if we want [math]\varphi[/math] to be faithful and positive:

[[math]] \varphi(f)=\int_Xf(x)d\mu(x) [[/math]]

In general, the positive linear forms can be thought of as being integration functionals with respect to some underlying “positive measures”. We can use them as follows:

Proposition

Let [math]\varphi:A\to\mathbb C[/math] be a positive linear form.

• [math] \lt a,b \gt =\varphi(ab^*)[/math] defines a generalized scalar product on [math]A[/math].
• By separating and completing we obtain a Hilbert space [math]H[/math].
• [math]\pi(a):b\to ab[/math] defines a representation [math]\pi:A\to B(H)[/math].
• If [math]\varphi[/math] is faithful in the above sense, then [math]\pi[/math] is faithful.

Show Proof

Almost everything here is straightforward, as follows:

(1) This is clear from definitions, and from the basic properties of the positive elements [math]a\geq0[/math], which can be established exactly as in the [math]A=B(H)[/math] case.

(2) This is a standard procedure, which works for any scalar product, the idea being that of dividing by the vectors satisfying [math] \lt x,x \gt =0[/math], then completing.

(3) All the verifications here are standard algebraic computations, in analogy with what we have seen many times, for multiplication operators, or group algebras.

(4) Assuming that we have [math]a\neq0[/math], we have then [math]\pi(aa^*)\neq0[/math], which in turn implies by faithfulness that we have [math]\pi(a)\neq0[/math], which gives the result.

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In order to establish the embedding theorem, it remains to prove that any [math]C^*[/math]-algebra has a faithful positive linear form [math]\varphi:A\to\mathbb C[/math]. This is something more technical:

Proposition

Let [math]A[/math] be a [math]C^*[/math]-algebra.

• Any positive linear form [math]\varphi:A\to\mathbb C[/math] is continuous.
• A linear form [math]\varphi[/math] is positive iff there is a norm one [math]h\in A_+[/math] such that [math]||\varphi||=\varphi(h)[/math].
• For any [math]a\in A[/math] there exists a positive norm one form [math]\varphi[/math] such that [math]\varphi(aa^*)=||a||^2[/math].
• If [math]A[/math] is separable there is a faithful positive form [math]\varphi:A\to\mathbb C[/math].

Show Proof

The proof here is quite technical, inspired from the existence proof of the probability measures on abstract compact spaces, the idea being as follows:

(1) This follows from Proposition 7.15, via the following estimate:

[[math]] |\varphi(a)| \leq||\pi(a)||\varphi(1) \leq||a||\varphi(1) [[/math]]

(2) In one sense we can take [math]h=1[/math]. Conversely, let [math]a\in A_+[/math], [math]||a||\leq1[/math]. We have:

[[math]] |\varphi(h)-\varphi(a)| \leq||\varphi||\cdot||h-a|| \leq\varphi(h) [[/math]]

Thus we have [math]Re(\varphi(a))\geq0[/math], and with [math]a=1-h[/math] we obtain:

[[math]] Re(\varphi(1-h))\geq0 [[/math]]

Thus [math]Re(\varphi(1))\geq||\varphi||[/math], and so [math]\varphi(1)=||\varphi||[/math], so we can assume [math]h=1[/math]. Now observe that for any self-adjoint element [math]a[/math], and any [math]t\in\mathbb R[/math] we have, with [math]\varphi(a)=x+iy[/math]:

[[math]] \begin{eqnarray*} \varphi(1)^2(1+t^2||a||^2) &\geq&\varphi(1)^2||1+t^2a^2||\\ &=&||\varphi||^2\cdot||1+ita||^2\\ &\geq&|\varphi(1+ita)|^2\\ &=&|\varphi(1)-ty+itx|\\ &\geq&(\varphi(1)-ty)^2 \end{eqnarray*} [[/math]]

Thus we have [math]y=0[/math], and this finishes the proof of our remaining claim.

(3) We can set [math]\varphi(\lambda aa^*)=\lambda||a||^2[/math] on the linear space spanned by [math]aa^*[/math], then extend this functional by Hahn-Banach, to the whole [math]A[/math]. The positivity follows from (2).

(4) This is standard, by starting with a dense sequence [math](a_n)[/math], and taking the Cesàro limit of the functionals constructed in (3). We have [math]\varphi(aa^*) \gt 0[/math], and we are done.

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With these ingredients in hand, we can now state and prove:

Theorem

Any [math]C^*[/math]-algebra appears as a norm closed [math]*[/math]-algebra of operators

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

over a certain Hilbert space [math]H[/math]. When [math]A[/math] is separable, [math]H[/math] can be taken to be separable.

Show Proof

This result, called called GNS representation theorem after Gelfand, Naimark and Segal, follows indeed by combining Proposition 7.15 with Proposition 7.16.

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All this might seem quite surprising, and your first reaction would be to say what have we been we doing here, with our [math]C^*[/math]-algebra theory, because we are now back to operator algebras [math]A\subset B(H)[/math], and everything that we did with [math]C^*[/math]-algebras, extending things that we knew about operator algebras [math]A\subset B(H)[/math], looks more like a waste of time.

Error. The axioms in Definition 7.6, coupled with the writing [math]A=C(X)[/math] in Definition 7.10, are something powerful, because they do not involve any kind of [math]L^2[/math] or [math]L^\infty[/math] functions on our quantum spaces [math]X[/math]. Thus, we can start hunting for such spaces, just by defining [math]C^*[/math]-algebras with generators and relations, then look for Haar measures on such spaces, and use the GNS construction in order to reach to von Neumann algebras. Before getting into this, however, let us summarize the above discussion as follows:

Theorem

We can talk about compact quantum measured spaces, as follows:

• The category of compact quantum measured spaces [math](X,\mu)[/math] is the category of the [math]C^*[/math]-algebras with faithful traces [math](A,\varphi)[/math], with the arrows reversed.
• In the case where we have a non-faithful trace [math]\varphi[/math], we can still talk about the corresponding space [math](X,\mu)[/math], by performing the GNS construction.
• By taking the weak closure in the GNS representation, we obtain the von Neumann algebra [math]A''=L^\infty(X)[/math], in the previous general measured space sense.

Show Proof

All this follows from Theorem 7.17, and from the other things that we already know, with the whole result itself being something rather philosophical.

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General references

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