Topological K-theory of C*-algebras for the Working Mathematician – Lecture 3 (Topological K-theory and three big theorems)

by Orr Shalit

Here is a write up of the third lecture. (Here are links to the first and second ones.) I want to stress that although Haim is giving me a lot of support in preparing these notes (thanks!), any mistakes you find here are my own.

In this lecture we briefly heard about the origin of K-theory (topological K-theory) and then we learned about three theorems (of Connes, Pimsner-Voiculescu and Schochet) describing how to compute the K-theory of various C*-algebras constructed from given C*-algebras in a given way.

1. Topological K-theory

K-theory for C*-algebras evolved from topological K-theory, which came first, and is still very important today. Here we will very quickly mention what it is. K-theory was first developed first in the 1950s by Grothendieck for algebraic geometry, but was not easy to work with. In 1961 Atiyah and Hirzebruch introduced K-theory for compact spaces (in this paper).

Recall that a vector bundle over a topological space X is a topological space E together with a continuous surjection \pi : E \rightarrow X such that \pi^{-1}(\{x\}) is a (complex) vector space for all x \in X, such that for every point x \in X there is a neighbourhood U \ni x such that \pi^{-1}(U) “looks like” U \times \mathbb{C}^n.

There is a natural addition of vector bundles, and therefore the collection of isomorphism classes of vector bundles forms an abelian group V(X). Therefore one may define

K^0(X) = \mathcal{G}(V(X)),

where \mathcal{G} denotes the Grothendiek construction of enveloping group.

It can be shown that K^0(X) = K_0(C(X)) – where the latter denotes the K_0 group of the C*-algebra C(X).


1) K^0(pt) = \mathbb{Z} – this is easy to see because the vector bundles over a point correspond to vector spaces, therefore the semigroup corresponding to vector bundles is \mathbb{N}.

2) K^0(\mathbb{R}P^4) = \mathbb{Z} \oplus \mathbb{Z}/4. Besides showing that there are non-trivial K-groups, this example shows that K-theory provides different information than ordinary cohomology with integer coefficients (H^{ev}(\mathbb{R}P^4, \mathbb{Z}) = \mathbb{Z} \oplus \mathbb{Z} / 2 \oplus \mathbb{Z} /2).

Atiyah and Janich showed that K^0(X) = [X, \mathcal{F}] – the homotopy classes of maps from X into the Fredholm operators \mathcal{F}. (A definition of Fredholm operators will be given in the next lecture).

2. Theorem of Connes

The first big theorem we discuss is Connes’s “analogue of the Thom isomorphism” (see here). Haim mentioned in class that the analogy between this result and the Thom isomorphism is not very revealing.

Connes’s theorem deals with computing the K-theory of the crossed product A \rtimes \mathbb{R}, where \mathbb{R} acts on A by an (point norm continuous) automorphism group \{\alpha_t\}_{t \in \mathbb{R}}.

Theorem: K_{j+1}(A \rtimes \mathbb{R}) \cong K_j(A).

One cannot ask for more.

Example: If \alpha is the trivial action, then A \rtimes \mathbb{R} = A \otimes C^*_r(\mathbb{R}) = A \otimes C_0(\mathbb{R}), which we recognise as the suspension SA. Thus

K_j(SA) = K_{j+1}(A),

which is expected for any homology theory (see the previous lecture).

A key idea in the proof of Connes’s theorem is Takai duality. Given a l.c. group abelian group action \alpha : G \rightarrow Aut(A),  then A \rtimes_\alpha G has a natural action \hat{\alpha} of \hat{G} on it. It is defined as follows: if f \in C(G,A) is a continuous and compactly supported A valued function on G, we define

\hat{\alpha}_\chi (f) (g) = \chi(g) f(g).

This extends to L^1(G) and hence to C^*(G), which is in this case equal to C^*_r(G).

Now we may form again the crossed product, and it turns out that we almost return to where we started:

(A \rtimes_\alpha G) \rtimes_{\hat{\alpha}} \hat{G} = A \otimes K.

The reason that we are not exactly where we started is not only that the C*-algebra changed from A to the Morita equivalent A \otimes K: the action \hat{\hat{\alpha}} is not merely \alpha \otimes id. In fact, K should be understood as the compact operators on L^2(G), and the the action is given by \alpha \otimes Ad(\lambda), where \lambda is the left regular representation.

 In his paper Connes used his theorem to deduce the following theorem, which was obtained earlier by Pimsner and Voiculescu.

3. Theorem of Pimsner-Voiculescu

Suppose that now we have a C*-algebra A, together with an action of \mathbb{Z}. In other words, we are given \alpha \in Aut(A). The following theorem related the K-theory of A \rtimes_\alpha \mathbb{Z} with that of A.

Theorem: There exists a six term exact sequence:

K_0(A) \xrightarrow{1-\alpha_*} K_0(A) \xrightarrow{i_*} K_0(A \rtimes \mathbb{Z})

\uparrow                                            \downarrow

K_1(A \rtimes \mathbb{Z}) \xleftarrow{i_*} K_1(A) \xleftarrow{1-\alpha_*} K_1 (A)

(Here, i denotes the inclusion of A into A \rtimes \mathbb{Z}.)

Example: The rotation algebra A_\lambda (Section 7 in lecture 1) is isomorphic to the reduced crossed product C(\mathbb{S}^1) \rtimes \mathbb{Z}, where \mathbb{Z} acts by rotating by angle \lambda. Thus, by PV exact sequence, and using K_0(C(\mathbb{S}^1)) = K_1(C(\mathbb{S}^1)) = \mathbb{Z}, we get the exact sequence

\mathbb{Z} \xrightarrow{0} \mathbb{Z} \longrightarrow K_0(A_\lambda)

\uparrow                                \downarrow

K_1(A_\lambda) \xleftarrow{i_*} \mathbb{Z} \xleftarrow{0} \mathbb{Z}

It follows that K_*(A_\lambda) = \mathbb{Z} \oplus \mathbb{Z}.

Example: Another nice example of a use of the PV six term exact sequence is the following. Let \mathbb{F}_n denote the free group on n generators. For a while it had been an open question whether C^*(\mathbb{F}_n) \cong C^*(\mathbb{F}_m) for m \neq n. Pimsner and Voiculescu used their six term exact sequence to show that K_0(\mathbb{F}_n) = \mathbb{Z} and that K_1(\mathbb{Z}) = \mathbb{Z}^{n+1}, settling this problem.

Example: As a final example, we mention that there was an open question due to Kaplansky whether there exists a simple unital C*-algebra with no nontrivial projection. This was answered positively by Blackadar, who was able to cook up an example of such a C*-algebra. A more natural example was provided by Connes, who (using K-theory) showed that if \phi is a minimal diffeomorphism of \mathbb{S}^3, then

A = C(\mathbb{S}^3) \rtimes_\phi \mathbb{Z}

is simple and has no nontrivial projections.

4. “Kunneth Theorem” of Schochet

We now arrive at Schochet’s “Kunneth Theorem” (from this paper) on how to compute the K-theory of tensor products.

Definition: The bootstrap category \mathcal{N} is the smallest category of separable nuclear C*-algebras that contains the complex numbers, is closed under direct limits, is closed under quotients (if A and J \lhd A are in \mathcal{N} then so is A/J), is closed under extensions (if J and A/J are in \mathcal{N} then so is A, and also if A and A/J are in \mathcal{N} then so is J), is closed under crossed products by \mathbb{R} and \mathbb{Z}, and is closed under KK-equivalence (which we have not defined).

It is a longstanding open problem whether the bootstrap category is equal to the category of separable nuclear C*-algebras. One reason why the answer to this open problem is very interesting is the following theorem.

Theorem: If A \in \mathcal{N} and B is a C*-algebra, then there exists a natural exact sequence

0 \rightarrow K_*(A) \otimes K_*(B) \rightarrow K_*(A \otimes B) \rightarrow Tor^1(K_*A, K_* B) \rightarrow 0

which splits unnaturally.