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 is a topological space together with a continuous surjection such that is a (complex) vector space for all , such that for every point there is a neighbourhood such that “looks like” .
There is a natural addition of vector bundles, and therefore the collection of isomorphism classes of vector bundles forms an abelian group . Therefore one may define
where denotes the Grothendiek construction of enveloping group.
It can be shown that – where the latter denotes the group of the C*-algebra .
1) – this is easy to see because the vector bundles over a point correspond to vector spaces, therefore the semigroup corresponding to vector bundles is .
2) . Besides showing that there are non-trivial K-groups, this example shows that K-theory provides different information than ordinary cohomology with integer coefficients ().
Atiyah and Janich showed that – the homotopy classes of maps from into the Fredholm operators . (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 , where acts on by an (point norm continuous) automorphism group .
One cannot ask for more.
Example: If is the trivial action, then , which we recognise as the suspension . Thus
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 , then has a natural action of on it. It is defined as follows: if is a continuous and compactly supported valued function on , we define
This extends to and hence to , which is in this case equal to .
Now we may form again the crossed product, and it turns out that we almost return to where we started:
The reason that we are not exactly where we started is not only that the C*-algebra changed from to the Morita equivalent : the action is not merely . In fact, should be understood as the compact operators on , and the the action is given by , where 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 , together with an action of . In other words, we are given . The following theorem related the K-theory of with that of .
Theorem: There exists a six term exact sequence:
(Here, denotes the inclusion of into .)
Example: The rotation algebra (Section 7 in lecture 1) is isomorphic to the reduced crossed product , where acts by rotating by angle . Thus, by PV exact sequence, and using , we get the exact sequence
It follows that .
Example: Another nice example of a use of the PV six term exact sequence is the following. Let denote the free group on generators. For a while it had been an open question whether for . Pimsner and Voiculescu used their six term exact sequence to show that and that , 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 is a minimal diffeomorphism of , then
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 is the smallest category of separable nuclear C*-algebras that contains the complex numbers, is closed under direct limits, is closed under quotients (if and are in then so is ), is closed under extensions (if and are in then so is , and also if and are in then so is ), is closed under crossed products by and , 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 and is a C*-algebra, then there exists a natural exact sequence
which splits unnaturally.