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
(*) $latex A_1 xrightarrow{f_1} A_2 xrightarrow{f_2} A_3 xrightarrow{f_3} ldots $
be a sequence of C*-algebras and maps. The direct limit $latex lim_{rightarrow} A_j$ 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
$latex lim_{alg} A_j = {a = (a_1, a_2, ldots ) in Pi A_j : exists n . forall j geq n . f_j (a_j) = a_{j+1}}.$
( The notation $latex lim_{alg}$ is temporary, not used beyond this section.) Because C*-maps are contractive, one has that $latex |a_{j+1}| leq |a_j|$ for all sufficiently large $latex j$, hence we may define $latex rho(a) = lim_j |a_j|$. Then the quotient space $latex lim_{alg} A_j / rho^{-1}(0)$ is a normed *-algebra, and completing it we get a C*-algebra denoted $latex lim_{rightarrow}A_j$ which is called the direct limit of the sequence (*). In the special case where every $latex f_j$ is isometric, then one often makes the identification
$latex A_1 subseteq A_2 subseteq A_3 subseteq ldots$
and the direct limit can be interpreted as the closure of the union $latex overline{cup A_j}$.
The maps $latex f_j$ promote to maps $latex iota_j : A_j rightarrow lim_{rightarrow} A_j$ such that $latex iota_j = iota_{j+1} circ f_j$ for all $latex j$. The C*-algebra $latex lim_{rightarrow}A_j$ together with the family of maps $latex {iota_j}$ has the following universal property:
Universal property for the direct limits. If $latex B$ is a C*-algebra, and if for every $latex j$ there is a map $latex g_j : A_j rightarrow B$ such that $latex g_j = g_{j+1}circ f_j$ for all $latex j$, then there is a unique map $latex f : lim_{rightarrow}A_j rightarrow B$ such that $latex f circ f_j = g_j$ for all $latex j$.
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 $latex mathcal{C}$. (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 $latex mathcal{C}$ be a category of C*-algebras and *-homomorphisms as maps ($latex mathcal{C}$ 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 $latex Ab$ the category of abelian groups with homomorphisms.
Definition: A homology theory for is a sequence $latex h_* = {h_n}_{n}$ of functors $latex h_n : mathcal{C} rightarrow Ab$ (indexed by $latex mathbb{Z}$ or by $latex mathbb{N}$) such that the following hold
- Homotopy axiom. If $latex f^0$ is homotopic to $latex f^1$ (see Section 5 here) then $latex f^0_* = f^1_*$. Here and below we shall denote by $latex f_*$ the map obtained by applying one of the functors on $latex f$, without being fussy about which $latex n$ we used.
- Exactness axiom. If $latex 0 rightarrow J xrightarrow{i} A xrightarrow{j} A/J rightarrow 0$ is a short exact sequence (s.e.s), then there is a natural long exact sequence (l.e.s.)
$latex ldots rightarrow h_n J xrightarrow{i_*} h_n A xrightarrow{j_*} h_n A/J xrightarrow{partial} h_{n-1}J rightarrow ldots $
Another way of stating this is to say that there exists a sequence of “connecting maps” $latex partial$ 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 $latex {A_j}$ of C*-algebras, the natural map is an isomorphism of $latex oplus_j h_n(A_j)$ onto $latex h_n(oplus_j A_j)$, for all $latex n$;
- Morita invariant if $latex h_n(A otimes K) cong h_n(A)$, or $latex h_n (A otimes M_2) cong h_n(A)$ for every $latex A$ and every $latex n$ (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 $latex A$ and $latex B$ are said to be stably isomorphic if their stabilisations $latex A otimes K$ and $latex B otimes K$ are isomorphic. There is another equivalence relation for C*-algebras – Morita Equivalence – and a theorem of Brown, Green and Rieffel says that two $latex sigma$ 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 $latex oplus_{jin J} G_j$ of a family of abelian groups $latex {G_j}_{j in J}$ is the set of sequences $latex (g_j)_{jin J}$ (where $latex g_j in G_j$) for which all but finitely many $latex g_j$s are $latex 0$ with coordinate-wise operations. The direct sum $latex oplus_{jin J} A_j$ of a family of C*-algebras $latex {A_j}_{j in J}$ is formed by first constructing the normed *-algebra of all sequences $latex (a_j)_{jin J}$ (where $latex a_j in A_j$) 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 $latex c_0$ direct sum”. The object that one might refer to as “$latex ell^infty$ direct sum” is usually referred to as “direct product”. )
Here are some quickly deducible consequences of the two axioms above. Suppose that $latex h_*$ is a homology theory.
Proposition 1: $latex h_*$ is finitely additive.
Proof. We have the following s.e.s. which splits on the left and on the right
$latex 0 rightarrow A_1 xrightarrow{leftarrow} A_1 oplus A_2 xrightarrow{leftarrow} A_2 rightarrow 0$.
From properties of functors, the long exact sequence also splits:
$latex ldots xrightarrow{partial} h_n(A_1) xrightarrow{leftarrow} h_n(A_1 oplus A_2) xrightarrow{leftarrow} h_n(A_2) xrightarrow{partial} h_{n-1}(A_1) rightarrow ldots $.
But then the connecting maps $latex partial$ are zero, and we have
$latex 0 rightarrow h_n(A_1) xrightarrow{leftarrow} h_n(A_1 oplus A_2) xrightarrow{leftarrow} h_n(A_2) rightarrow 0$
so $latex h_n(A_1 oplus A_2) cong h_n(A_1) oplus h_n(A_2)$.
Proposition 2: $latex h_*(CA) = 0$.
Proof. $latex CA$ is contractible (recall Sections 3.5 and 5 here for notation).
Proposition 3: $latex h_n(A) = h_{n-1}(SA)$.
Proof. We have the natural exact sequence
$latex 0 rightarrow SA rightarrow CA rightarrow A rightarrow 0$
where $latex CA rightarrow A$ is given by evaluation at $latex 1$. Then the l.e.s. gives
$latex 0 = h_n(CA) rightarrow h_n(A) rightarrow h_{n-1}(SA) rightarrow h_{n-1}(CA) = 0$,
so $latex partial$ is an isomorphism.
Remark: If $latex h_*$ is a (additive) homology theory and $latex N$ is a nuclear C*-algebra then $latex A to h_*(A otimes N)$ is a (additive) homology theory.
We will require the following theorem.
Theorem (Theorem 5.1 in the paper): Let $latex h_*$ is an additive homology theory. Let $latex A = lim_{rightarrow} A_j$ be the direct limit of the sequence
$latex A_1 xrightarrow{f_1} A_2 xrightarrow{f_2} ldots $.
Then for all $latex n$ the maps $latex h_n(A_j) rightarrow h_n(A)$ induce an isomorphism $latex lim_{rightarrow} h_n(A_j) = h_n(A)$.
3. The mapping cone and the Meyer-Vietoris Theorem
Let $latex A xrightarrow{f} B$. A pull back of $latex f$ is an algebra $latex P$ together with maps $latex P rightarrow A$ and $latex P rightarrow CB$ that complete the top left corner of the following diagram and make it commutative:
$latex P longrightarrow A$
$latex downarrow $ $latex downarrow$
$latex CB longrightarrow B$
This completion problem can always be solved as follows. Define the mapping cone of $latex f$ to be
$latex C_f = {(xi,a) in CB oplus A mid xi(1) = f(a) }$.
Why would we want to introduce the algebra $latex C_f$? Suppose we want to show that $latex h_n A = h_n B$. Then having introduced, the mapping cone, we have the s.e.s.
$latex 0 rightarrow SB rightarrow C_f rightarrow A rightarrow 0, $
which by the exactness axiom gives the l.e.s.
$latex h_n(C_f) rightarrow h_n(A) rightarrow h_{n-1}(SB) rightarrow h_{n-1}(C_f)$
$latex searrow$ $latex ||$
$latex h_n(B)$
and the big diagram commutes. Thus to show that $latex h_n A = h_n B$ (by the natural map) it is necessary and sufficient to show that $latex h_n(C_f) = h_{n-1} = 0$.
The Meyer-Vietoris Theorem. Let $latex P xrightarrow{g_i} A_i$ be a pullback of the two surjective maps $latex A_i xrightarrow{f_i} B$,
$latex P longrightarrow A_1$
$latex downarrow $ $latex downarrow$
$latex A_2 longrightarrow B$.
Then there is a l.e.s.
$latex h_ P xrightarrow{(g_{1*},g_{2*})} h_n A_1 oplus h_n A_2 xrightarrow{(-f_{1*},f_{2*})} h_n B rightarrow h_{n-1}P $.
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 $latex A otimes K$ of a C*-algebra $latex A$. However, suppose that we have defined K-theory for unital C*-algebras. Then for every C*-algebra we let $latex A^+ mapsto A^+ / A cong mathbb{C}$ be the natural quotient of the unitalization of $latex A$ by $latex A$, and we can then define $latex K_*(A)$ to be the kernel of the map $latex K_*(A^+) rightarrow K_*(mathbb{C})$. (This will be consistent in case that $latex A$ is unital to begin with). Thus, for the definitions of K-theory we stick with unital algebras.
Purely ring theoretic definition. Let $latex mathcal{R}$ be the set of all equivalence classes of finitely generated projective left $latex A$-modules. Define an addition on $latex mathcal{R}$ by $latex [P_1] + [P_2] = [P_1 oplus P_2]$. This makes $latex mathcal{R}$ into a abelian semigroup with $latex 0$. From any abelian semigroup $latex mathcal{S}$ with $latex 0$ one may form a group $latex mathcal{G}(mathcal{S})$ containing it, called the Grothendiek group; roughly, it is the set of all formal differences $latex s_1 – s_2$. Then $latex K_0(A)$ is defined to be $latex mathcal{G}(mathcal{R})$.
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., $latex su = tu Rightarrow s = t$). For the (possibly) non cancellative semigroups arising in K-theory, two formal differences $latex [P_1] – [P_2] $ and $latex [Q_1] – [Q_2]$ are considered as the same point if and only if there exists a finitely generated projective $latex mathcal{R}$-module $latex M$ such that $latex [P_1] + [Q_2] + [M] = [Q_1] + [P_2] + [M]$.
We have only defined $latex K_0(A)$, but by Proposition 3 above, if $latex K_*$ is a homology theory then $latex K_1(A) cong K_0(SA)$, $latex K_2(A) cong K_1(SA)$, etc. Thus one really only needs to give a definition of $latex K_0$. (To be precise you only obtain positive integer indexed groups this way. In the end there will be only $latex K_0$ and $latex K_1$ so this doesn’t matter).
Example: If $latex A = mathbb{C}$, then a finitely generated projective module over $latex A$ is just a complex vector space, and the equivalence classes can be identified with the natural numbers $latex mathbb{N}$, where every $latex n$ represents the equivalence class of $latex n$ dimensional vector spaces. Addition is easily seen to correspond to addition of dimension, so $latex mathcal{R}$ here is also, as a semigroup, equal to $latex mathbb{N}$. Now $latex K_0(mathbb{C}) = mathcal{G}(mathbb{N}) = mathbb{Z}$. Note that here the semigroup $latex mathbb{N}$ is cancellative, so $latex mathcal{G}(mathbb{N})$ really does look like the set of formal differences $latex mathbb{N} – mathbb{N}$.
Matrix definition. Let $latex M_n(A)$ denote the matrix algebra over $latex A$. The observation that a finitely generated projective left $latex A$-module leads to the following definition of $latex K_0$.
Denote
$latex mathcal{P}_n A = { p in M_n(A) : p = p^* = p^2 }. $
We embed $latex mathcal{P}_ nA$ in $latex mathcal{P}_{n+1}A$ by
$latex p mapsto left(begin{smallmatrix} p & 0 \ 0 & 0 end{smallmatrix}right) .$
The we let $latex mathcal{P}_infty A$ be the set of equivalence classes of projections in $latex cup_n mathcal{P}_n A$, for the equivalence relation $latex p sim q$ if and only if $latex p$ is unitarily equivalent to $latex q$ in some $latex M_n(A)$. We let addition on $latex mathcal{P}_n(A)$ be defined by
$latex [p] + [q] = left(begin{smallmatrix} p & 0 \ 0 & q end{smallmatrix}right) .$
Finally, we let $latex K_0(A) = mathcal{G}(mathcal{P}_infty A)$.
Example: Using this definition of $latex K_0$, it is easy to recalculate $latex K_0(mathbb{C}) = mathbb{Z}$, since in this case $latex mathcal{P}_infty A cong mathbb{N}$.
Example: Here is a good example of why the Grothendieck construction is in general not just “formal differences”. Consider $latex A = B(H)$ (with $latex H$ separable infinite dimensional). We calculate $latex mathcal{P}_infty A cong mathbb{N} cup {infty}$, where $latex infty = [I]$ corresponds to projections of infinite rank. Then for every $latex p,q in M_n (A)$, we have $latex [p]+ [I] = [q] + [I]$. It follows that in $latex mathcal{G}(mathcal{P}_infty A)$ all elements are equivalent, so $latex K_0(B(H)) = 0$.
Even though we remarked above that in principle one need only define $latex K_0$, it is interesting and useful to give a direct definition of $latex K_1(A)$.
Denote by $latex mathcal{U}_nA$ the set of all unitaries in $latex M_n(A)$. Embed $latex mathcal{U}_n A $in $latex mathcal{U}_{n+1} A$ by
$latex u mapsto left(begin{smallmatrix} u & 0 \ 0 & 1 end{smallmatrix}right) .$
We the let $latex mathcal{U}_infty A$ be the increasing union of all $latex mathcal{U}_n A$, and say that $latex u sim v$ if they are unitarily equivalent in $latex mathcal{U}_n A$ (Recall that the identifications that we have made, this means that $latex diag(u,1)$ and $latex diag(v,1)$ are unitarily equivalent, where $latex u$ and $latex v$ 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 $latex K_1(A)$ be $latex mathcal{U}_infty(A)$.
There are some things to prove here. For example, it is not clear why $latex K_1(A)$ must be an Abelian group (but it is). Also, it is not on the surface, but it is true, that if $latex u$ and $latex v$ are path wise connected in $latex mathcal{U}_n A$, then $latex u sim v$. Thus we have
Example: $latex K_1(mathbb{C}) = 0$.
(Because $latex mathcal{U}_n mathbb{C}$ is path connected.)
With the above two definitions of $latex K_0$ and $latex K_1$, it is easy how to define what the functor does to maps: every $latex A xrightarrow{f} B$ promotes to a map, also denoted $latex f$ between the matrix algebras $latex M_n(A)$ and $latex M_n(B)$, which is also a C*-map hence sends projections to projections, and when $latex f$ is unital it sends unitaries to unitaries. Then we define $latex f_* : K_*(A) rightarrow K_*(B)$ by
$latex f_* [a] = [f(a)]$.
Homotopy definition: The following definition also works for every $latex n geq 1$:
$latex K_n(A) = pi_{n-1}(mathcal{U}_infty A)$,
where $latex pi_{k}$ denotes the $latex k$th homotopy group.
This definition shows that for $latex K_1$ we could have used the invertibles $latex GL_n(A) subset M_n(A)$ rather than the unitaries $latex mathcal{U}_n A$.
5. The Main Theorem
Theorem: $latex K_*$ is an additive and Morita invariant homology theory. Moreover it is $latex 2$-periodic, i.e., $latex K_n(S^2 A) = K_n(A)$ (Bott Periodicity). Equivalently, the l.e.s. becomes a cyclic six term exact sequence:
$latex K_1(J) rightarrow K_1(A) rightarrow K_1(A/J)$
$latex uparrow$ $latex downarrow$
$latex K_0(A/J) leftarrow K_0(A) leftarrow K_0(J) $
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 $latex K_1(A/J) xrightarrow{partial} K_0(J)$ 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 $latex K_0(A/J) rightarrow K_1(J)$ is the trickier one.
Here is the definition of $latex K_1(A/J) xrightarrow{partial} K_0(J)$. If $latex b in GL_n(A/J)$ then $latex left[ begin{smallmatrix} b & 0\ 0 & b^{-1} end{smallmatrix} right] in GL_{2n}(A/J)$. Let $latex a in GL_{2n}(A)$ be a lift of $latex left[ begin{smallmatrix} b & 0 \ 0 & b^{-1} end{smallmatrix} right]$. Then define
$latex partial [b] = [a p_n a^{-1}] – [p_n]$,
where $latex p_n$ is the projection on the first $latex n$ coordinates along the diagonal.
Example: This example should explain the terminology “index map”. Consider the s.e.s. $latex 0 rightarrow K rightarrow B(H) rightarrow Q rightarrow 0$. A unitary in $latex u in Q$ is the image of a Fredholm partial isometry $latex v in B(H)$, that is a partial isometry with finite kernel and finite cokernel. Then
$latex left[ begin{smallmatrix} u & 0 \ 0 & u^{-1} end{smallmatrix}right]$
lifts to $latex left[ begin{smallmatrix} v & 1 – vv^* \ 1 – v^* v & v^* end{smallmatrix} right]$ .
A calculation shows that using the definition of $latex partial$ just given, we have
$latex partial [u] = $
$latex left[ begin{smallmatrix} v v^* & 0 \ 0 & 1-v^*v end{smallmatrix} right]$ $latex – left[begin{smallmatrix} 1&0 \ 0&0 end{smallmatrix} right]$
and this element in $latex K_0(K) = mathbb{Z}$ is the equivalence class of in $latex [1-v^* v] – [1 – v v^*]$, which corresponds to the difference between the dimensions of these projections, i.e., the Fredholm index of $latex v$.
6. Some examples
We now examine several examples, using the Main Theorem from Section 2.
Examples:
1) $latex K_*(M_n) = K_*(mathcal{K}) = K_*(mathbb{C}$, which is $latex mathbb{Z}$ (for $latex K_0$) and $latex 0$ (f0r $latex K_1$). This follows from Morita invariance.
2) Consider the sequence $latex M_2 rightarrow M_4 rightarrow M_8 rightarrow M_{16} rightarrow cdots$, and let $latex A = lim_{rightarrow} A_j$. The K-theory of $latex A$ (and therefore also its isomorphism class) depends on the connecting maps.
If the maps are
$latex a mapsto left[begin{smallmatrix} a & 0 \ 0 & 0 end{smallmatrix} right]$
The the induced sequence $latex K_0(M_2) rightarrow K_0(M_4) rightarrow K_0(M_8) rightarrow cdots$ becomes
$latex mathbb{Z} xrightarrow{1} mathbb{Z} xrightarrow{1} mathbb{Z} rightarrow cdots$,
which is the constant sequence with identity maps. Thus $latex K_0(A) = mathbb{Z}$.
On the other hand, if
$latex a mapsto left[begin{smallmatrix} a & 0 \ 0 & a end{smallmatrix} right]$
then on the level K groups we get
$latex mathbb{Z} xrightarrow{2} mathbb{Z} xrightarrow{2} mathbb{Z} rightarrow cdots$,
and the inductive limit os then $latex K_0(A) = lim_{rightarrow} K_0(A_j) = mathbb{Z}[1/2]$.
If one takes a sequence of appropriately sized matrix algebras, and at each step mapa $latex a$ to $latex diag(a,a,ldots,a)$, the one may obtain any countable abelian group. In particular, taking a sequence of inflations
$latex 2,3,2,3,5,2,3,5,7,2,3,5,7,11,ldots$
then for the direct limit algebra $latex A$ will have $latex K_0(A) = mathbb{Q}$.
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