Topological K-theory of C*-algebras for the Working Mathematician – Lecture 2 (Definitions and core examples)
by Orr Shalit
This is a write-up of the second lecture in the course given by Haim Schochet. For the first lecture and explanations, see the previous post.
I will very soon figure out how to put various references online and post links to that, too.
1. Direct limits
We began the lecture with an important construction: the direct limit (also called inductive limit) of a sequence of C*-algebras. Let
be a sequence of C*-algebras and maps. The direct limit is formed as follows (the direct limit depends also on the maps, but these are usually omitted from the notation and understood). First, form the space
( The notation is temporary, not used beyond this section.) Because C*-maps are contractive, one has that for all sufficiently large , hence we may define . Then the quotient space is a normed *-algebra, and completing it we get a C*-algebra denoted which is called the direct limit of the sequence (*). In the special case where every is isometric, then one often makes the identification
and the direct limit can be interpreted as the closure of the union .
The maps promote to maps such that for all . The C*-algebra together with the family of maps has the following universal property:
Universal property for the direct limits. If is a C*-algebra, and if for every there is a map such that for all , then there is a unique map such that for all .
If one wants to work with inverse limits of C*-algebras, then things don’t work so well, and one has to work with pro-C*-algebras. This is closely related to the fact that the direct limit of a sequence of locally compact spaces need not be locally compact.
There is also a related notion of direct limit in the category of groups (or in other well behaved categories). It is simpler than the direct limit of C*-algebras, since one does not have to define a norm and complete. (The direct limit of a group is just the group of all eventually coherent sequences, modulo the relation of being eventually equal.)
2. Homology theories for C*-algebras
There are several different ways to define homology (or cohomology) theories for topological spaces. Topologists never expect to have a homology theory defined for all topological spaces. K-theory is rather special in that it is a homology theory that makes sense and is well behaved for the category of all C*-algebras . (Haim says that this is just “dumb luck”.)
K-theory is a homology theory for C*-algebras. Before defining K-theory, let’s see what is a homology theory. A reference for this section is the paper “Topological methods for C*-algebras. III. Axiomatic Homology“, by C. Schochet.
Let be a category of C*-algebras and *-homomorphisms as maps ( can be taken to be the full category of C*-algebras, but sometimes we’ll stick to a subcategory, e.g., separable, nuclear, etc.). We denote by the category of abelian groups with homomorphisms.
Definition: A homology theory for is a sequence of functors (indexed by or by ) such that the following hold
- Homotopy axiom. If is homotopic to (see Section 5 here) then . Here and below we shall denote by the map obtained by applying one of the functors on , without being fussy about which we used.
- Exactness axiom. If is a short exact sequence (s.e.s), then there is a natural long exact sequence (l.e.s.)
Another way of stating this is to say that there exists a sequence of “connecting maps” which make the above l.e.s. exact.
In addition to the above two basic axioms, there are additional desirable axioms that we may wish for (but we will not assume). For example, a homology is said to be:
- Additive (or countably additive, if one wants to emphasise) if for every sequence of C*-algebras, the natural map is an isomorphism of onto , for all ;
- Morita invariant if , or for every and every (continuity properties of a homology theory will show that the two different assumptions are equivalent).
It will turn out that K-theory satisfies these two additional axioms. The Morita invariance axiom can be stated as the fact that stably isomorphic C*-algebras have the same homology. Recall that two C*-algebras and are said to be stably isomorphic if their stabilisations and are isomorphic. There is another equivalence relation for C*-algebras – Morita Equivalence – and a theorem of Brown, Green and Rieffel says that two unital C*-algebras are stably isomorphic if and only if they are Morita equivalent. Hence the terminology. (A reference for the last paragraph is Chapter 7 in E. Lance’s monograph on Hilbert C*-modules.)
(Parenthetical remark: It may be worth clarifying what we mean by direct sum. The direct sum of a family of abelian groups is the set of sequences (where ) for which all but finitely many s are with coordinate-wise operations. The direct sum of a family of C*-algebras is formed by first constructing the normed *-algebra of all sequences (where ) with coordinate-wise operations and sup norm, and then completing it with respect to that norm. Thus, the direct sum of C*-algebras is what one sometimes calls “the direct sum”. The object that one might refer to as “ direct sum” is usually referred to as “direct product”. )
Here are some quickly deducible consequences of the two axioms above. Suppose that is a homology theory.
Proposition 1: is finitely additive.
Proof. We have the following s.e.s. which splits on the left and on the right
From properties of functors, the long exact sequence also splits:
But then the connecting maps are zero, and we have
Proposition 2: .
Proof. is contractible (recall Sections 3.5 and 5 here for notation).
Proposition 3: .
Proof. We have the natural exact sequence
where is given by evaluation at . Then the l.e.s. gives
so is an isomorphism.
Remark: If is a (additive) homology theory and is a nuclear C*-algebra then is a (additive) homology theory.
We will require the following theorem.
Theorem (Theorem 5.1 in the paper): Let is an additive homology theory. Let be the direct limit of the sequence
Then for all the maps induce an isomorphism .
3. The mapping cone and the Meyer-Vietoris Theorem
Let . A pull back of is an algebra together with maps and that complete the top left corner of the following diagram and make it commutative:
This completion problem can always be solved as follows. Define the mapping cone of to be
Why would we want to introduce the algebra ? Suppose we want to show that . Then having introduced, the mapping cone, we have the s.e.s.
which by the exactness axiom gives the l.e.s.
and the big diagram commutes. Thus to show that (by the natural map) it is necessary and sufficient to show that .
The Meyer-Vietoris Theorem. Let be a pullback of the two surjective maps ,
Then there is a l.e.s.
In fact, it is enough to assume that only one of the maps is surjective.
4. Definition(s) of K-theory
It is very important to be able to treat non-unital algebras in K-theory. For example, one wants to consider the stabilisation of a C*-algebra . However, suppose that we have defined K-theory for unital C*-algebras. Then for every C*-algebra we let be the natural quotient of the unitalization of by , and we can then define to be the kernel of the map . (This will be consistent in case that is unital to begin with). Thus, for the definitions of K-theory we stick with unital algebras.
Purely ring theoretic definition. Let be the set of all equivalence classes of finitely generated projective left -modules. Define an addition on by . This makes into a abelian semigroup with . From any abelian semigroup with one may form a group containing it, called the Grothendiek group; roughly, it is the set of all formal differences . Then is defined to be .
One should be a little more careful, though: the Grothendiek group of a semigroup can be identified with formal differences only if the semigroup has cancellation (i.e., ). For the (possibly) non cancellative semigroups arising in K-theory, two formal differences and are considered as the same point if and only if there exists a finitely generated projective -module such that .
We have only defined , but by Proposition 3 above, if is a homology theory then , , etc. Thus one really only needs to give a definition of . (To be precise you only obtain positive integer indexed groups this way. In the end there will be only and so this doesn’t matter).
Example: If , then a finitely generated projective module over is just a complex vector space, and the equivalence classes can be identified with the natural numbers , where every represents the equivalence class of dimensional vector spaces. Addition is easily seen to correspond to addition of dimension, so here is also, as a semigroup, equal to . Now . Note that here the semigroup is cancellative, so really does look like the set of formal differences .
Matrix definition. Let denote the matrix algebra over . The observation that a finitely generated projective left -module leads to the following definition of .
We embed in by
The we let be the set of equivalence classes of projections in , for the equivalence relation if and only if is unitarily equivalent to in some . We let addition on be defined by
Finally, we let .
Example: Using this definition of , it is easy to recalculate , since in this case .
Example: Here is a good example of why the Grothendieck construction is in general not just “formal differences”. Consider (with separable infinite dimensional). We calculate , where corresponds to projections of infinite rank. Then for every , we have . It follows that in all elements are equivalent, so .
Even though we remarked above that in principle one need only define , it is interesting and useful to give a direct definition of .
Denote by the set of all unitaries in . Embed in by
We the let be the increasing union of all , and say that if they are unitarily equivalent in (Recall that the identifications that we have made, this means that and are unitarily equivalent, where and may or may not have been of the same size, and one adds as many ones on the diagonal as needed to make this unitary equivalence work). Finally, define be .
There are some things to prove here. For example, it is not clear why must be an Abelian group (but it is). Also, it is not on the surface, but it is true, that if and are path wise connected in , then . Thus we have
(Because is path connected.)
With the above two definitions of and , it is easy how to define what the functor does to maps: every promotes to a map, also denoted between the matrix algebras and , which is also a C*-map hence sends projections to projections, and when is unital it sends unitaries to unitaries. Then we define by
Homotopy definition: The following definition also works for every :
where denotes the th homotopy group.
This definition shows that for we could have used the invertibles rather than the unitaries .
5. The Main Theorem
Theorem: is an additive and Morita invariant homology theory. Moreover it is -periodic, i.e., (Bott Periodicity). Equivalently, the l.e.s. becomes a cyclic six term exact sequence:
A few words instead of a proof. A proof will not be presented here, rather this mini-course will aim at advanced application, using theorems such as this as a black box (see Blackadar’s book, Chapter 4 Section 9). The hardest part of this theorem is the Bott periodicity. The map is called the index map. The tricky part in the proof of this theorem is the definition of the connecting maps and the verification of exactness at the edges of these maps. Of the two maps, the map is the trickier one.
Here is the definition of . If then . Let be a lift of . Then define
where is the projection on the first coordinates along the diagonal.
Example: This example should explain the terminology “index map”. Consider the s.e.s. . A unitary in is the image of a Fredholm partial isometry , that is a partial isometry with finite kernel and finite cokernel. Then
lifts to .
A calculation shows that using the definition of just given, we have
and this element in is the equivalence class of in , which corresponds to the difference between the dimensions of these projections, i.e., the Fredholm index of .
6. Some examples
We now examine several examples, using the Main Theorem from Section 2.
1) , which is (for ) and (f0r ). This follows from Morita invariance.
2) Consider the sequence , and let . The K-theory of (and therefore also its isomorphism class) depends on the connecting maps.
If the maps are
The the induced sequence becomes
which is the constant sequence with identity maps. Thus .
On the other hand, if
then on the level K groups we get
and the inductive limit os then .
If one takes a sequence of appropriately sized matrix algebras, and at each step mapa to , the one may obtain any countable abelian group. In particular, taking a sequence of inflations
then for the direct limit algebra will have .