1b. Gelfand theorem

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In this section we discuss the two main results regarding the [math]C^*[/math]-algebras. First we have the Gelfand theorem, which is particularly interesting for us, in view of our quantum space and quantum group motivations. Then we have the GNS representation theorem, that we will use less often, but which is something fundamental too.

The Gelfand theorem, which will be fundamental for us, is as follows:

Theorem (Gelfand)

Any commutative [math]C^*[/math]-algebra is the form

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

with its “spectrum” [math]X=Spec(A)[/math] appearing as the space of characters [math]\chi :A\to\mathbb C[/math].

Show Proof

Given a commutative [math]C^*[/math]-algebra [math]A[/math], we can define indeed [math]X[/math] to be the set of 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]. Then [math]X[/math] is a compact space, and [math]a\to ev_a[/math] is a morphism of algebras:

[[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{i(a-a^*)}{2} [[/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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The Gelfand theorem has some important philosophical consequences. Indeed, in view of this theorem, we can formulate the following 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, in rigorous mathematical parlance, we can define the category of the compact quantum spaces to be the category of the [math]C^*[/math]-algebras, with the arrows reversed. And that's all. QED.

When [math]A[/math] is commutative, the space [math]X[/math] considered above 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.

Less enthusiastically now, we will see a bit later, after developing some more theory, that this formalism has its limitations, and needs a fix. But more on this later.

As a first consequence of the Gelfand theorem, we can extend Theorem 1.8 above to the case of the normal elements ([math]aa^*=a^*a[/math]), in the following way:

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 can develop as well the theory of positive elements, in analogy with the theory of positive operators, as follows:

Theorem

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 something very standard, as follows:

[math](1)\implies(2)[/math] This follows from Proposition 1.12, because we can use the function [math]f(z)=\sqrt{z}[/math], which is well-defined on [math]\sigma(a)\subset[0,\infty)[/math], and so set [math]b=\sqrt{a}[/math].

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

[math](2)\implies(1)[/math] Observe that this is clear too, because we have:

[[math]] \begin{eqnarray*} \sigma(a) &=&\sigma(b^2)\\ &=&\sigma(b)^2\\ &\subset&[0,\infty) \end{eqnarray*} [[/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]. But this contradicts the elementary fact that [math]\sigma(dd^*),\sigma(d^*d)[/math] must coincide outside [math]\{0\}[/math], coming from Proposition 1.7 above.

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Let us review now the other fundamental result regarding the [math]C^*[/math]-algebras, namely the representation theorem of Gelfand, Naimark and Segal. We first 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 an embedding

[[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 [math]f\in C(X)[/math], consider the following operator, on the space [math]H=L^2(X)[/math]:

[[math]] T_f(g)=fg [[/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 general, the idea will be that of extending this construction. We will need:

Definition

Consider a linear map [math]\varphi:A\to\mathbb C[/math].

• [math]\varphi[/math] is called positive when [math]a\geq0\implies\varphi(a)\geq0[/math].
• [math]\varphi[/math] is called faithful and positive when [math]a \gt 0\implies\varphi(a) \gt 0[/math].

In the commutative case, [math]A=C(X)[/math], the positive linear forms 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 Theorem 1.13.

(2) This is a standard procedure, which works for any scalar product.

(3) All the verifications here are standard algebraic computations.

(4) This follows indeed from [math]a\neq0\implies\pi(aa^*)\neq0\implies\pi(a)\neq0[/math].

■

In order to establish the GNS theorem, it remains to prove that any [math]C^*[/math]-algebra has a faithful and 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, which is quite technical, inspired from the existence proof of the probability measures on abstract compact spaces, goes as follows:

(1) This follows from Proposition 1.16, via the following inequality:

[[math]] \begin{eqnarray*} |\varphi(a)| &\leq&||\pi(a)||\varphi(1)\\ &\leq&||a||\varphi(1) \end{eqnarray*} [[/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]] \begin{eqnarray*} |\varphi(h)-\varphi(a)| &\leq&||\varphi||\cdot||h-a||\\ &\leq&\varphi(h)1\\ &=&\varphi(h) \end{eqnarray*} [[/math]]

Thus we have [math]Re(\varphi(a))\geq0[/math], and it remains to prove that the following holds:

[[math]] a=a^*\implies\varphi(a)\in\mathbb R [[/math]]

By using [math]1-h\geq 0[/math] we can apply the above to [math]a=1-h[/math] and we obtain:

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

We conclude that [math]Re(\varphi(1))\geq Re(\varphi(h))=||\varphi||[/math], and so [math]\varphi(1)=||\varphi||[/math]. Summing up, 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 the following inequality:

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

On the other hand with [math]\varphi(a)=x+iy[/math] we have:

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

We therefore obtain that for any [math]t\in\mathbb R[/math] we have:

[[math]] \varphi(1)^2(1+t^2||a||^2)\geq(\varphi(1)-ty)^2 [[/math]]

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

(3) Consider the linear subspace of [math]A[/math] spanned by the element [math]aa^*[/math]. We can define here a linear form by the following formula:

[[math]] \varphi(\lambda aa^*)=\lambda||a||^2 [[/math]]

This linear form has norm one, and by Hahn-Banach we get a norm one extension to the whole [math]A[/math]. The positivity of [math]\varphi[/math] follows from (2).

(4) Let [math](a_n)[/math] be a dense sequence inside [math]A[/math]. For any [math]n[/math] we can construct as in (3) a positive form satisfying [math]\varphi_n(a_na_n^*)=||a_n||^2[/math], and then define [math]\varphi[/math] in the following way:

[[math]] \varphi=\sum_{n=1}^\infty\frac{\varphi_n}{2^n} [[/math]]

Let [math]a\in A[/math] be a nonzero element. Pick [math]a_n[/math] close to [math]a[/math] and consider the pair [math](H,\pi)[/math] associated to the pair [math](A,\varphi_n)[/math], as in Proposition 1.16. We have then:

[[math]] \begin{eqnarray*} \varphi_n(aa^*) &=&||\pi(a)1||\\ &\geq&||\pi (a_n)1||-||a-a_n||\\ &=&||a_n||-||a-a_n||\\ & \gt &0 \end{eqnarray*} [[/math]]

Thus [math]\varphi_n(aa^*) \gt 0[/math]. It follows that 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 (GNS theorem)

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

• [math]A[/math] appears as a closed [math]*[/math]-subalgebra [math]A\subset B(H)[/math], for some Hilbert space [math]H[/math].
• When [math]A[/math] is separable (usually the case), [math]H[/math] can be chosen to be separable.
• When [math]A[/math] is finite dimensional, [math]H[/math] can be chosen to be finite dimensional.

Show Proof

This result follows indeed by combining the construction from Proposition 1.16 above with the existence result from Proposition 1.17.

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Generally speaking, the GNS theorem is something very powerful and concrete, which perfectly complements the Gelfand theorem, and the resulting compact quantum space formalism. We can always go back to good old Hilbert spaces, whenever we get lost.

So long for linear operators, operator algebras, and quantum spaces. The above material, mixing ideas from math and physics, algebra and analysis, and so on, might seem quite wizarding, and indeed it is. This is subtle, first class mathematical physics, taking a long time to be fully understood, and here is some suggested further reading:

(1) Mathematically speaking, all the above, while certainly tricky, and most of what will follow too, is formally based on 3rd year mathematics as we know it, meaning Rudin . However, for better understanding all this, get to learn some more functional analysis, operator theory, and basic operator algebras, say from Lax .

(2) In what regards operator algebras, that we will heavily use in what follows, some more knowledge would be welcome too. Be aware tough that most books here are deeply committed to physics of the 1920s, which is not enough for our purposes here. A good, very useful book is Blackadar . And for more, go with Connes .

(3) Importantly now, while you will certainly survive all that follows only knowing Rudin , you will not survive it without some physics knowledge. Believe me. Why bothering with quantum spaces, or with quantum groups, or with mathematics, or with life in general. Without some clear motivations, this ain't going anywhere.

(4) The standard place for learning physics are the books of Feynman , , . If you already know some physics, you can try as well Griffiths , , , equally fun, and a bit more advanced, and quantum mechanics oriented. And if looking for a more compact package, two concise books, go with Weinberg , .

(5) As a crucial piece of advice now, physics is a whole, and you won't get away just by reading some quantum mechanics. Depending on your choice between Feynman, Griffiths, Weinberg, don't hesitate to complete with more classical mechanics, from Kibble or Arnold , and more thermodynamics, from Schroeder or Huang .

And this is, I guess, all you need to know. By the way, in relation with all this physics, if looking for something really compact, assuming all basic mathematics known, but basic physics unknown, you can try as well my book . Although, as for any physics book written by a mathematician, that might contain severe physical mistakes.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].