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This chapter is a continuation of the previous one, meant to be a grand finale to the [math]C^*[/math]-algebra theory that we started to develop there, before getting back to more traditional von Neumann algebra material, following Murray, von Neumann and others. There are countless things to be said, and possible paths to be taken. En hommage to Connes, and his book , which is probably the finest ever on [math]C^*[/math]-algebras, we will adopt a geometric viewpoint. To be more precise, we know that a [math]C^*[/math]-algebra is a beast of type [math]A=C(X)[/math], with [math]X[/math] being a compact quantum space. So, it is about the “geometry” of [math]X[/math] that we would like to talk about, everything else being rather of administrative nature.

Let us first look at the classical case, where [math]X[/math] is a usual compact space. You might say right away that wrong way, what we need for doing geometry is a manifold. But my answer here is modesty, and no hurry. It is right that you cannot do much geometry with a compact space [math]X[/math], but you can do some, and we have here, for instance:

Definition

Given a compact space [math]X[/math], its first [math]K[/math]-theory group [math]K_0(X)[/math] is the group of formal differences of complex vector bundles over [math]X[/math].

This notion is quite interesting, and we can talk in fact about higher [math]K[/math]-theory groups [math]K_n(X)[/math] as well, and all this is related to the homotopy groups [math]\pi_n(X)[/math] too. There are many non-trivial results on the subject, the end of the game being of course that of understanding the “shape” of [math]X[/math], that you need to know a bit about, before getting into serious geometry, in the case where [math]X[/math] happens to be a manifold.

As a question for us now, operator algebra theorists, we have: \begin{question} Can we talk about the first [math]K[/math]-theory group [math]K_0(X)[/math] of a compact quantum space [math]X[/math]? \end{question} We will see that this is a quite subtle question. To be more precise, we will see that we can talk, in a quite straightforward way, of the group [math]K_0(A)[/math] of an arbitrary [math]C^*[/math]-algebra [math]A[/math], which is constructed as to have [math]K_0(A)=K_0(X)[/math] in the commutative case, where [math]A=C(X)[/math], with [math]X[/math] being a usual compact space. In the noncommutative case, however, [math]K_0(A)[/math] will sometimes depend on the choice of [math]A[/math] satisfying [math]A=C(X)[/math], and so all this will eventually lead to a sort of dead end, and to a rather “no” answer to Question 8.2.

Getting started now, in order to talk about the first [math]K[/math]-theory group [math]K_0(A)[/math] of an arbitrary [math]C^*[/math]-algebra [math]A[/math], we will need the following simple fact:

Proposition

Given a [math]C^*[/math]-algebra [math]A[/math], the finitely generated projective [math]A[/math]-modules [math]E[/math] appear via quotient maps [math]f:A^n\to E[/math], so are of the form

[[math]] E=pA^n [[/math]]

with [math]p\in M_n(A)[/math] being an idempotent. In the commutative case, [math]A=C(X)[/math] with [math]X[/math] classical, these [math]A[/math]-modules consist of sections of the complex vector bundles over [math]X[/math].

Show Proof

Here the first assertion is clear from definitions, via some standard algebra, and the second assertion is clear from definitions too, again via some algebra.

■

With this in hand, let us go back to Definition 8.1. Given a compact space [math]X[/math], it is now clear that its [math]K[/math]-theory group [math]K_0(X)[/math] can be recaptured from the knowledge of the associated [math]C^*[/math]-algebra [math]A=C(X)[/math], and to be more precise we have [math]K_0(X)=K_0(A)[/math], when the first [math]K[/math]-theory group of an arbitrary [math]C^*[/math]-algebra is constructed as follows:

Definition

The first [math]K[/math]-theory group of a [math]C^*[/math]-algebra [math]A[/math] is the group of formal differences

[[math]] K_0(A)=\big\{p-q\big\} [[/math]]

of equivalence classes of projections [math]p\in M_n(A)[/math], with the equivalence being given by

[[math]] p\sim q\iff\exists u, uu^*=p,u^*u=q [[/math]]

and with the additive structure being the obvious one, by diagonal concatenation.

This is very nice, and as a first example, we have [math]K_0(\mathbb C)=\mathbb Z[/math]. More generally, as already mentioned above, it follows from Proposition 8.3 that in the commutative case, where [math]A=C(X)[/math] with [math]X[/math] being a compact space, we have [math]K_0(A)=K_0(X)[/math]. Observe also that we have, by definition, the following formula, valid for any [math]n\in\mathbb N[/math]:

[[math]] K_0(A)=K_0(M_n(A)) [[/math]]

Some further elementary observations include the fact that [math]K_0[/math] behaves well with respect to direct sums and with inductive limits, and also that [math]K_0[/math] is a homotopy invariant, and for details here, we refer to any introductory book on the subject, such as .

In what concerns us, back to our Question 8.2, what has been said above is certainly not enough for investigating our question, and we need more examples. However, these examples are not easy to find, and for getting them, we need more theory. We have:

Definition

The second [math]K[/math]-theory group of a [math]C^*[/math]-algebra [math]A[/math] is the group of connected components of the unitary group of [math]GL_\infty(A)[/math], with

[[math]] GL_n(A)\subset GL_{n+1}(A)\quad,\quad a\to\begin{pmatrix}a&0\\0&1\end{pmatrix} [[/math]]

being the embeddings producing the inductive limit [math]GL_\infty(A)[/math].

Again, for a basic example we can take [math]A=\mathbb C[/math], and we have here [math]K_1(\mathbb C)=\{1\}[/math], trivially. In fact, in the commutative case, where [math]A=C(X)[/math], with [math]X[/math] being a usual compact space, it is possible to establish a formula of type [math]K_1(A)=K_1(X)[/math]. Further elementary observations include the fact that [math]K_1[/math] behaves well with respect to direct sums and with inductive limits, and also that [math]K_1[/math] is a homotopy invariant.

Importantly, the first and second [math]K[/math]-theory groups are related, as follows:

Theorem

Given a [math]C^*[/math]-algebra [math]A[/math], we have isomorphisms as follows, with

[[math]] SA=\left\{f\in C([0,1],A)\Big|f(0)=0\right\} [[/math]]

standing for the suspension operation for the [math]C^*[/math]-algebras:

• [math]K_1(A)=K_0(SA)[/math].
• [math]K_0(A)=K_1(SA)[/math].

Show Proof

Here the isomorphism in (1) is something rather elementary, and the isomorphism in (2) is something more complicated. In both cases, the idea is to start first with the commutative case, where [math]A=C(X)[/math] with [math]X[/math] being a compact space, and understand there the isomorphisms (1,2), called Bott periodicity isomorphisms. Then, with this understood, the extension to the general [math]C^*[/math]-algebra case is quite straightforward.

■

The above result is quite interesting, making it clear that the groups [math]K_0,K_1[/math] are of the same nature. In fact, it is possible to be a bit more abstract here, and talk in various clever ways about the higher [math]K[/math]-theory groups, [math]K_n(A)[/math] with [math]n\in\mathbb N[/math], of an arbitrary [math]C^*[/math]-algebra, with the result that these higher [math]K[/math]-theory groups are subject to Bott periodicity:

[[math]] K_n(A)=K_{n+2}(A) [[/math]]

However, in practice, this leads us back to Definition 8.4, Definition 8.5 and Theorem 8.6, with these statements containing in fact all we need to know, at [math]n=0,1[/math].

Going ahead with examples, following Cuntz and related papers, we have:

Theorem

The [math]K[/math]-theory groups of the Cuntz algebra [math]O_n[/math] are given by

[[math]] K_0(O_n)=\mathbb Z_{n-1}\quad,\quad K_1(O_n)=\{1\} [[/math]]

with the equivalent projections [math]P_i=S_iS_i^*[/math] standing for the standard generator of [math]\mathbb Z_{n-1}[/math].

Show Proof

We recall that the Cuntz algebra [math]O_n[/math] is generated by isometries [math]S_1,\ldots,S_n[/math] satisfying [math]S_1S_1^*+\ldots+S_nS_n^*=1[/math]. Since we have [math]S_i^*S_i=1[/math], with [math]P_i=S_iS_i^*[/math], we have:

[[math]] P_1\sim\ldots\sim P_n\sim 1 [[/math]]

On the other hand, we also know that we have [math]P_1+\ldots+P_n=1[/math], and the conclusion is that, in the first [math]K[/math]-theory group [math]K_1(O_n)[/math], the following equality happens:

[[math]] n[1]=[1] [[/math]]

Thus [math](n-1)[1]=0[/math], and it is quite elementary to prove that [math]k[1]=0[/math] happens in fact precisely when [math]k[/math] is a multiple of [math]n-1[/math]. Thus, we have a group embedding, as follows:

[[math]] \mathbb Z_{n-1}\subset K_0(O_n) [[/math]]

The whole point now is that of proving that this group embedding is an isomorphism, which in practice amounts in proving that any projection in [math]O_n[/math] is equivalent to a sum of the form [math]P_1+\ldots+P_k[/math], with [math]P_i=S_iS_i^*[/math] as above. Which is something non-trivial, requiring the use of Bott periodicity, and the consideration of the second [math]K[/math]-theory group [math]K_1(O_n)[/math] as well, and for details here, we refer to Cuntz and related papers.

■

The above result is very interesting, for various reasons. First, it shows that the structure of the first [math]K[/math]-theory groups [math]K_0(A)[/math] of the arbitrary [math]C^*[/math]-algebras can be more complicated than that of the first [math]K[/math]-theory groups [math]K_0(X)[/math] of the usual compact spaces [math]X[/math], with the group [math]K_0(A)[/math] being for instance not ordered, in the case [math]A=O_n[/math], and with this being the first in a series of no-go observations that can be formulated.

Second, and on a positive note now, what we have in Theorem 8.7 is a true noncommutative computation, dealing with an algebra which is rather of “free” type. The outcome of the computation is something nice and clear, suggesting that, modulo the small technical issues mentioned above, we are on our way of developing a nice theory, and that the answer to Question 8.2 might be “yes”. However, as bad news, we have:

Theorem

There are discrete groups [math]\Gamma[/math] having the property that the projection

[[math]] \pi:C^*(\Gamma)\to C^*_{red}(\Gamma) [[/math]]

is not an isomorphism, at the level of [math]K[/math]-theory groups.

Show Proof

For constructing such a counterexample, the group [math]\Gamma[/math] must be definitely non-amenable, and the first thought goes to the free group [math]F_2[/math]. But it is possible to prove that [math]F_2[/math] is [math]K[/math]-amenable, in the sense that [math]\pi[/math] is an isomorphism at the [math]K[/math]-theory level. However, counterexamples do exist, such as the infinite groups [math]\Gamma[/math] having Kazhdan's property [math](T)[/math]. Indeed, for such a group the asssociated Kazhdan projection [math]p\in K_0(C^*(\Gamma))[/math] is nonzero, while mapping to the zero element [math]0\in K_0(C^*_{red}(\Gamma))[/math], so we have our counterexample.

■

As a conclusion to all this, which might seem a bit dissapointing, we have: \begin{conclusion} The answer to Question 8.2 is no. \end{conclusion} Of course, the answer to Question 8.2 remains “yes” in many cases, the general idea being that, as long as we don't get too far away from the classical case, the answer remains “yes”, so we can talk about the [math]K[/math]-theory groups of our compact quantum spaces [math]X[/math], and also, about countless other invariants inspired from the classical theory. For a survey of what can be done here, including applications too, we refer to Connes' book .

In what concerns us, however, we will not take this path. For various reasons, coming from certain quantum physics beliefs, which can be informally summarized as “at sufficiently tiny scales, freeness rules”, we will be rather interested in this book in compact quantum spaces [math]X[/math] which are of “free” type, and we will only accept geometric invariants for them which are well-defined. And [math]K[/math]-theory, unfortunately, does not qualify.

General references

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