### Topological K-theory of C*-algebras for the Working Mathematician – Lecture 4 (K-homology and Brown-Douglas-Fillmore)

My notes of Haim’s Schochet’s fourth lecture in this series is here below.

It is impossible to start without mention that Alexander Grothendieck passed away last week. Grothendieck is considered by many as one of the greatest mathematicians of 20th century, and his contributions affect the material in this lecture series in at least two significant ways. As we mentioned, a first version of K-theory was developed by Grothendieck opening the door for topological K-theory (which, in turn, opened the door for K-theory of C*-algebras). Grothendieck also developed the theory of nuclear topological vector spaces and tensor products of topological vector spaces, a theory that has influenced the development of the concepts of nuclearity and tensor product which are central to contemporary C*-algebra theory.

#### 1. The homology theory associated with K-theory

In the previous lecture we discussed topological K-theory (of topological spaces), that is the functor $X \mapsto K(X)$. One can define $K^0(X) = K(X)$ and $K^n(X) = K(S^n X)$ and then $K^*$ is a cohomology theory, and in fact $K^*(X) = K_*(C(X))$ (the latter denotes the K-theory of the C*-algebra $C(X)$).

Now, topologists who are well versed in general nonsense know that every cohomology theory has a homology theory with which it is associated (assuming that we are working in a well behaved class of topological spaces. For the present section we assume all our topological spaces are finite CW complexes). What is the homology theory associated with K-theory?

One way is to define it using Spanier-Whitehead duality. If $X$ is a finite CW complex, then one can always find a big enough $n$ such that $X$ embeds in the northern hemisphere of the sphere $S^{2n+1}$, avoiding the north pole. Let $DX$ denote the complement of $X$ in $S^{2n+1}$. One may deformation-retract $DX$ to the southern hemisphere (avoiding the south pole, for technical reasons). Then $DX$ is called the Spanier-Whitehead dual of $X$. Different ways of embedding $X$ in $S^{2n+1}$ (in particular different choices of $n$) may lead to a different topological space $DX$, but $DX$ is determined uniquely up to stable homotopy.

A theorem (of Spanier and Whitehead? Anderson? Ask Haim) asserts that the regular homology and cohomology of $X$ and $DX$ are related by

$H_*(X) \cong H^{*+k}(DX)$

and

$H_*(DX) \cong H^{*+k}(X)$

for some $k$.

Coming back to the question “what is the homology theory associated with K-theory?”, we see that we can try to define $K_n(X) = K^n(DX)$. This does work, and defines a homology theory, but a deficiency of this approach is that it only works for finite CW complexes, and does not work for general compact spaces (as K-theory did).

#### 2. Fredholm operators

Fredolm operator is a bounded operator $T \in B(X,Y)$ between two Banach spaces, such that $ker T$  and $coker T := Y / Im T$ are finite dimensional (one usually throws in the assumption that $T$ has closed range, but this follows automatically if I am not mistaken). For such an operator it makes sense to define the index of $T$, defined by

$ind(T) = \dim ker T - \dim Y / Im T.$

We will be mostly concerned with the case where $X = Y = H$ a Hilbert space, and then a Fredholm operator can be defined to be an operator $T \in B(H)$ with closed range such that $\dim ker T, \dim ker T^* < \infty$ (in this formulation one has to include the assumption that the range is closed) and the index is then defined as

$ind(T) = \dim ker T - \dim ker T^*.$

Denote by $\mathcal{F} \subseteq B(X,Y)$ the set of Fredholm operators. Then operators in $\mathcal{F}$ have the following prperties

1. If $T \in \mathcal{F}$ and $A \in K(X,Y)$ is compact, then $T + A \in \mathcal{F}$ and $ind(T) = ind(T+A)$.
2. If $S,T \in \mathcal{F}$ (perhaps on different spaces) then $S T \in \mathcal{F}$ and $ind(ST) = ind(S) + ind(T)$.
3. $T \in \mathcal{F}$ if and only if $T$ is invertible modulo the compacts. In particular $\mathcal{F}$ is open.
4. If $T \in \mathcal{F}$ and $A$ has sufficiently small norm then $T +A \in \mathcal{F}$ and $ind(T) = ind(T+A)$. Thus the index is a continuous function $\mathcal{F} \rightarrow \mathbb{Z}$.

Fredholm operators are a very tractable class of operators, and they have applications to partial differential and to integral equations (in fact the theory was born with Fredholm’s solution of integral equations arising from potential theory). Only decades after they were introduced and extensively used was a connection made between Fredhom operators and topology. The connection was Atiyah and Singer’s index theorem.

#### 3. Atiyah-Singer index theorem and Atiyah’s question

In 1963 Atiyah and Singer proved their famous “Atiyah-Singer index theorem” (ASIT below), which was generalised in a series of papers during the 60s. I will make the long story very short, and the Wikipedia article I linked to above seems like an excellent place to start if you want to learn more.

The ASIT concerns elliptic operators on smooth manifolds. Elliptic operators are partial differential operators which have nice solvability properties similar to those of the Laplacian, and they are of importance in mathematical physics. A partial differential operator can be considered as a bounded operator, if interpreted as acting between the appropriate Soboloev spaces. In case the operator was elliptic, then it is not just bounded but also a Fredholm operator. Thus it has a Fredhom index, which in this setting is referred to as the analytical index.

There is another index associated with an elliptic partial differential operator an a smooth manifold called the topological index. This index is defined in terms of the partial differential operator and some other fancy topological data related to the manifold at hand.

The ASIT asserts that for an elliptic operator on a smooth manifold the topological and the analytical indices are the same. It is a far reading generalisation of many results relating the topological structure of a manifold with the dimensions of spaces of solutions of some operator, such as the classical Riemann-Roch theorem (the Riemann-Roch theorem determines the dimension of the space of meromorphic functions on a connected compact Riemann surface with given zeros and poles in terms of the genus of of the surface).

Haim told us in the lecture that according to Encyclopaedia Britannia, this theorem is one of the greatest mathematical achievements in the 1960s.

Elliptic operators are partial differential operators and are classically defined only on differentiable manifolds. Atiyah went one step further and asked for a definition of elliptic operators on a general compact space $X$. He argued that an elliptic operator should be defined as follows.

Suppose that we have a compact space $X$, a Hilbert space $H$, and a faithful unital representation $\sigma : C(X) \rightarrow B(H)$. Let us identify every $f \in C(X)$ with its image $\sigma(f)$ under this representation.

Definition: $T \in B(H)$ is elliptic if

1. $T$ is Fredholm, and
2. for all $f \in C(X)$, $[T,f] = Tf - fT$ is compact.

Denote by $\mathcal{E}ll(X)$ the set of all pairs $(\sigma,T)$ where $T$ is an elliptic operator relative to $\sigma$ as above.

Remark: A word of motivation is in order. What does this definition have to do with elliptic operators? The requirement that $T$ be Fredholm comes from the nice solvability properties of elliptic operators, namely, that they are Fredholm. The requirement $[T,f]$ is compact comes from the property of any differential operator $D$ of order $n$, that $[D,f]$ is an operator of order $n-1$ (this is easy to check formally for $f$ smooth). By the Sobolev embedding theorems this means that $[D,f]$ is compact if $D$ was bounded.

Using the index map and Spanier-Whitehead duality, Atiyah was able to define (in the case that $X$ is a finite CW complex) a surjective map

$\mathcal{E}ll(X) \rightarrow K^0(DX) = K_0(X)$.

And then (1969) Atiyah asked the following question.

Question: What equivalence relation on $\mathcal{E}ll(X)$ is needed to make this into an isomorphism?

This is a remarkable question, and if I am not mistaken it is the first instance of an attempt to construct a functor of topological using operator theory.

#### 4. The theory of Brown-Douglas-Fillmore

Answering Atiyah’s question would enable one to define the homology related to K-theory in operator theoretic terms. A partial answer to the above question was given by Brown, Douglas and Fillmore (BDF below) in the 1970s. They approached this from a totally different direction (see this paper. I will be glossing over many significant details). We need a few more definitions, and then we need to go further back in time in order to place the work of BDF in full context.

In what follows, $K$ denotes the compact operators on a Hilbert space $H$, and $Q$ denotes the Calkin algebra $B(H)/K$ (in the lectures as well as in their paper the Calkin algebra was denoted $\mathfrak{A}$, but I will stick with my $Q$ for typographical reasons). We use $\pi$ to denote the projection $\pi : B(H) \rightarrow Q$.

Definitions: Let $T \in B(H)$. The essential spectrum of $T$ is defined to be

$\sigma_e(T) = \sigma(\pi(T))$.

We say that an operator $T$ has some property essentially, if $\pi(T)$ has this property. Thus $T$ is essentially normal if $\pi(T)$ is normal, and $T_1,T_2$ essentially commute if $\pi(T_1), \pi(T_2)$ commute. One exception to this rule is that if $\pi(T)$ is invertible then $T$ is not called essentially invertible; it is simply called Fredholm.

The story of BDF does not start with topological homology theories, but with operator theory.

Weyl-von Neumann Theorem (1909, 1935): Let $A,B$ be two self adjoint operators on $H$. Then $\sigma_e(A) = \sigma_e(B)$ if and only if $A$ is unitarily equivalent to $B$ modulo the compacts (or: essentially unitarily equivalent). To be precise, there exists a unitary $U \in B(H)$ such that $A - U^* B U \in K$.

Remark: There is some ambiguity in saying that two operators $A,B \in B(H)$ are essentially unitarily equivalent (or unitarily equivalent modulo the compacts). One reasonable meaning is that there is a unitary $U \in B(H)$ such that $A - U^* B U \in K$. Another reasonable meaning is that there is a unitary $u \in Q$ such that $\pi (A) - u^* \pi(B) u = 0$. The point is that it is not every a unitary $u \in Q$ can be lifted to unitary $U \in B(H)$. We will take the first meaning. (BDF actually prove that these notions are equivalent, but this will not enter our discussion.)

The WvN theorem can be reformulated in the (seemingly slightly weaker form): For any two $a,b \in Q$ selfadjoint elements in the Calkin algebra, $a$ is unitarily equivalent to $b$ (in $Q$) if and only if $\sigma(a) = \sigma(b)$. (It is not hard to see that one may replace $A$ and $B$ above with essentially selfadjoint operators).

Thus, WvN theorem tells us that selfadjoint elements in $Q$ are classified by their spectrum. It is natural to seek an extension of this theorem to normal elements in $Q$. This leads us to examine essentially normal operators in $B(H)$, that is, operators $A$ such that $[A,A^*] \in K$. What do essentially normal operators look like?

Example: If $N$ is normal and $A$ is compact, then $N + A$ is essentially normal.

Do all essentially normal operators look lie this? No: $ind(N+A) = 0$ for operators as above, while on the other hand:

Example: Let $S: \ell^2(\mathbb{N}) \rightarrow \ell^2(\mathbb{N})$ be the unilateral shift

$S(x_1, x_2, x_3, \ldots) = (x_2, x_3, x_4, \ldots)$.

Then $S^* S - S S^*$ is a rank one projection, so $S$ is essentially normal.

The operator $S$ cannot be written as a sum of a normal plus a compact because

$ind(S) = \dim ker S - \dim ker S^* = -1$.

In fact, $\pi(S)$ is unitary and from symmetry consideration one obtains $\sigma(\pi(S)) = \mathbb{T}$. Thus the essential spectrum $\sigma_e(S)$ of $S$ is the unit circle.

Let $U : \ell^2(\mathbb{Z}) \rightarrow \ell^2(\mathbb{Z})$ be the bilateral shift

$S(\ldots,x_{-1},\underline{x_0}, x_1, \ldots) = (\ldots, x_0, \underline{x_1},x_2, \ldots)$.

Then $U$ is also essentially normal (it is normal) and its essential spectrum is also $\mathbb{T}$. However, $U$ and $S$ are not essentially unitarily equivalent, from the above considerations.

We see that a nonzero index is an obstruction for an essentially normal operator to be a normal plus compact. Is it the only obstruction? If two essentially normal operators have the same essential spectrum, is the index (where defined) the only obstruction for them to be essentially unitarily equivalent?

BDF’s approach to these problems initiated a paradigm shift in the operator theory from “single operator theory” to the theory of operator algebra.

For an essentially normal operator, BDF wrote down the exact sequence

$0 \rightarrow K \rightarrow C^*(I,K,T) \rightarrow C^*(1,\pi(T)) \rightarrow 0$,

where $C^*(I,K,T)$ denotes the C*-algebra generated by the identity, the compacts and $T$. Since $T$ is essentially normal the quotient C*-algebra $C^*(1,\pi(T))$ is commutative, and it is equal to $C(\sigma_e(T))$. BDF observed that the sequence splits if and only if $T$ is equal to a normal plus compact. From this observation they derived their approach to the problems above: classify the equivalence classes of exact sequences (“extensions of $C(X)$ by $K$“, or vice-versa, however you decide to call it) of the form

(*) $0 \rightarrow K \rightarrow E \rightarrow C(X) \rightarrow 0$

where $E \subseteq B(H)$ is a unital C*-algebra containing the compacts and $X$ is a compact metrisable space (for solving the operator theoretic problems above it was sufficient to consider only $X \subseteq \mathbb{C}$). Two extensions $E_1, E_2$ (of $C(X)$ by $K$) are said to be equivalent if there is an isomorphism $E_1 \rightarrow E_2$ that makes the diagram commute. Because of the representation theory of the compacts such an isomorphism must be implemented by a unitary in $B(H)$.

Note that one may alternatively think of the extension (*) as a monomorphism from $C(X)$ into $Q$, and seek to classify such monomorphisms. The trivial extension (one that splits) corresponds to a monomorphism that lifts through $B(H)$, and the two monomorphisms $\sigma_1, \sigma_2 : C(X) \rightarrow Q$ are equivalent if and only if they are unitarily equivalent by a unitary in $B(H)$, that is $\sigma(\cdot) = \pi(U) \sigma_2(\cdot) \pi(U)^*$ for a unitary in $B(H)$.

Note that two essentially normal operators with the same essential spectrum give rise to equivalent extensions if and only they are unitarily equivalent modulo the compacts.

Denote the set of all extensions as (*) by $Ext(X)$. Every extension (*) gives rise to an exact sequence in K-theory, and in particular we obtain the connecting map

$K^1(X) = K_1(C(X)) \xrightarrow{\gamma_\infty} K_0(\mathcal{K}) = \mathbb{Z}$,

and note that $\gamma_\infty \in Hom(K^1(X), \mathbb{Z})$.

BDF showed that $K^1(X)$ is isomorphic to the free abelian group on the connected components of $\mathbb{C} \setminus X$, and therefore $Hom(K^1(X) , \mathbb{Z}) \cong [\mathbb{C}\setminus X , \mathbb{Z}]$ – the continuous functions from the complement of $X$ to the integers. Note that whenever $T$ is essentially normal, the map $\lambda \mapsto ind(T-\lambda I)$ is a well defined continuous integer valued function on $\mathbb{C} \setminus \sigma_e(T)$.

Theorem (BDF):  For a compact subset $X \subset \mathbb{C}$, the map

$Ext(X) \rightarrow Hom(K^1(X), \mathbb{Z})$

given by

$[T] \mapsto (\lambda \mapsto ind(T-\lambda I))$

is an isomorphism.

An immediate consequence is the following solution to the problems we stated above regarding classification of essentially normal operators.

Theorem (BDF):

1. If $T_1,T_2$ are essentially normal and $\sigma_e(T_1) = \sigma_e(T_2) = X$ then $T_1$ is unitarily equivalent to $T_2$ modulo the compacts if and only if $ind(T_1 - \lambda I) = ind(T_2 - \lambda I)$ for all $\lambda \notin X$.
2. If $T$ is essentially normal, then it is of the form normal plus compact if and only if $in(T - \lambda I)$ for all $\lambda \notin \sigma_e(T)$.

#### 5. K-homology

Next, BDF set out to find how $Ext$ behaves as a functor. They replaced $C(X)$ by an arbitrary separable C*-algebra, and considered extensions of the form

$0 \rightarrow K \rightarrow E \rightarrow A \rightarrow 0$.

In modern notation, one defines $K^1(A) = Ext(A)$, and $K^0(A) = K^1(SA)$.

Then one has the following theorems.

Theorem (Bott periodicity): $K^*(A) = K^*(S^2A)$.

Theorem: $K^*$ is a functor form separable nuclear C*-algebras into graded abelian groups.

Theorem: $K^*$ is a cohomology theory on separable nuclear C*-algebras, which is homotopy invariant, additive, Morita invariant and satisfies the exactness axiom.

When restricting attention algebras of the form $A = C(X)$ one obtains a homology theory for spaces, which extends the homology theory given by Spanier-Whitehead duality that we described above. So BDF’s work gives us a definition of sought for homology theory in terms of operator algebras, and answers, in a way, the question asked by Atiyah. The homology theory coming from the $K^*$ groups is called K-homology. (It is somewhat unsuccessful terminology that we have on the cohomological side “K-theory”, and on the homological side “K-homology”. But that’s the way it is.)

Regarding direct limits, the situation is a little more complicated.

Theorem: If $A = \lim_{\rightarrow} A_j$ is a direct limit of C*-algebras. Then there is a short exact sequence of the form

$0 \rightarrow \lim^1 K^{*-1}(A_j) \rightarrow K^*(A_j) \rightarrow \lim_{\leftarrow} K^*(A_j) \rightarrow 0$.

The $\lim^1$ group is always either uncountable or $0$. Without going into details of what this means (honestly, I don’t understand this), we consider an example that illustrates some consequences.

Example: Consider the direct sequence

$M_2 \rightarrow M_4 \rightarrow M_8 \rightarrow \ldots$

with the maps

$x \mapsto \left[\begin{smallmatrix}x & 0 \\ 0 & x \end{smallmatrix}\right]$,

and denote the direct limit by $A$.

We have computed in the past that $K_0(A) = \mathbb{Z}[1/2]$ and $K_1(A) = 0$. On the other hand one can show that $K^0(A) = 0$ and that $K^1(A) = \hat{\mathbb{Z}}_2 / \mathbb{Z}$. Here $\hat{\mathbb{Z}}_2$ denotes the 2-adic integers, and is uncountable. Hence the set of equivalence classes of extensions (by the compacts) of the separable C*-algebra $A$ is uncountable.