Non-commutative Atiyah-Janich theorem

[This is the first of probably very many math related posts I’m going to make. I intend on posting notes for lectures and talks that I find either particularly interesting to me. Hopefully, they will be of interest to others as well. Some details are missing from this post and the output is pretty ugly. I’ll correct these when I stop being lazy. Thanks go out to Eusebio for giving an awesome talk.]

This post are the lecture notes from a seminar talk given by Eusebio Gardella of the same title, during the Analysis seminar at the University of Oregon on May 10, 2011.

Let $H$ be an infinite dimensional, separable Hilbert space. We denote the set of bounded linear operators of $H$ as $B(H)$. $B(H)$ has a two-sided closed ideal $K(H)$ consisting of the compact operators of $H$ (i.e. operators that map bounded sets into precompact sets). $K(H)$ is the closure of the finite rank operators, which forms a (not closed) two-sided ideal of $B(H)$. We start off with a well-known theorem that doubles as a definition.

Atkinson’s Theorem

For any bounded linear operator $F$ of $H$, the following are equivalent:

1. There exists a bounded linear operator $G$ of $H$ such that $1- FG$ and $1- GF$ are compact operators.
2. The image of $F$ in the quotient map $B(H)\rightarrow B(H)/K(H)$ is invertible.
3. $\ker\,F$, $\hbox{coker}\,F$ are finite dimensional and $\hbox{ran}\,F$ is closed.

It should be immediately clear that (1) and (2) are equivalent. If $F$ satisfies any of the conditions above, we say that $F$ is a Fredholm operator and the Fredholm index of $F$ is defined by

$\displaystyle\hbox{ind}\,F=\dim\ker\,F-\dim\hbox{coker}\,F\in\mathbb{Z}.$

If we take $H=\ell^2$ and define the unilateral shift $S$ by $Se_n=e_{n+1}$, where $e_n$ is the sequence with 1 in the $n$th coordinate and $0$ else. $S$ is injective, $\hbox{coker}\,S=\mathbb{C}e_1$ and $\hbox{ran}\,S=\ell^2(\mathbb{N}\setminus\{1\})$. So $S$ is a Fredholm operator and $\hbox{ind}\,S=-1$.

We denote $\hbox{Fred}(H)$ as the set of Fredholm operators of $H$. Note that $\hbox{Fred}(H)$ forms a semi-group under composition (but is not additively closed). We have the following properties of the index map:

1. $\hbox{ind}:\hbox{Fred}(H)\rightarrow\mathbb{Z}$ is locally constant.
2. $\hbox{ind}(F_1\circ F_2)=\hbox{ind}(F_1)+\hbox{ind}(F_2)$.
3. $\hbox{ind}(F+K)=\hbox{ind}(F)$ for all compact operators $K$.
4. If $\hbox{ind}(F)=0$, then $F+K$ is invertible for some compact operator $K$.
5. $\hbox{ind}(F^*)=-\hbox{ind}(F)$.

This last statement holds because $\hbox{coker}\,F=\ker\,F^*$.

Theorem 2

The index characterizes the connected components of $\hbox{Fred}(H)$, i.e. $\hbox{ind}(F_1)= \hbox{ind}(F_2)$ if and only if $F_1$ and $F_2$ are homotopic (i.e. in the same connected component).

This follows from the properties above and the fact that $\hbox{ind}^{-1}(0)$ is connected. This in turn follows from the fact that $\hbox{ind}^{-1}(0)=K(H)+GL(H)$, which are both connected.

So we now turn to the commutative version of the theorem that we will generalize.

Atiyah-Janich Theorem

Let $X$ be a compact, Hausdorff space. We denote $[X,\hbox{Fred}(H)]$ as the set of homotopy classes of continuous maps from $X$ into $\hbox{Fred}(H)$. There is a a group isomorphism

$\displaystyle \hbox{ind}: [X,\hbox{Fred}(H)]\rightarrow K^0(X).$

Note that if $X$ is just a point, then $K^0(X)=\mathbb{Z}$ and the index map is the Fredholm index. Also note that there is a natural isomorphism $K^0(X)\cong K_0(C(X))$.

Non-commutative Atiyah-Janich Theorem
Let $A$ be a $\sigma$-unital $C^*$-algebra. There is a group isomorphism
$\displaystyle \hbox{ind}:\pi_0(\hbox{Fred}(H_A))\rightarrow K_0(A).$

So what does all this mean? $K_0(A)$ is the Grothendieck group of $V(A)$, which is the set of isomorphism classes of finitely generated projective modules over $A$ with direct sum being the semi-group operation. We can think about $K_0(A)$ as the set of formal differences of elements in $V(A)$.

Note that in the case where $A=\mathbb{C}$, finitely generated projective modules over $A$ are the same as finite dimensional vector spaces. The isomorphism classes of vector spaces correspond to their dimension and the direct sum corresponds to addition of dimensions. So $V(A)\cong\mathbb{N}$ and so $K_0(A)\cong\mathbb{Z}$.

To describe $H_A$ requires some definitions. A right Hilbert $C^*$-module over $A$ is a triple $(X,\cdot,\langle,\rangle)$ consisting of a Banach space $X$, a continuous right action $\cdot: X\times A\rightarrow X$ and an inner product $\langle\cdot,\cdot\rangle:X\times X\rightarrow A$ such that some conditions are satisfied.

An example is if $I$ is an ideal of $A$, then $I$ is a $C^*$-module by the action being right multiplication and $\langle i,j\rangle =i^*j$.

Another example is $H_A=\ell^2\otimes A=\{(a_k)_{k\geq 1}\subseteq A: \sum_{k=1}^{\infty} a_k^*a_k \mbox{ converges in } A\}$. $(a_k)\cdot b=(a_kb)$ and $\langle (a_k), (b_k)\rangle = \sum_{k=1}^{\infty} a_k^*b_k$. This is known as the standard Hilbert $A$-module.

A map $T:X\rightarrow Y$ between Hilbert modules is said to be adjointable if there exists $T^*:Y\rightarrow X$ such that $\langle Tx,y\rangle = \langle x,T^*y\rangle$. We denote $\mathcal{L}(X,Y)$ the set of adjointable maps, we have a norm, which is reminiscent of the operator norm $\|T\|=\sup\{\|Tx\|: \|x\|=1\}$. Further, $\mathcal{L}(X)$ is a $C^*$-algebra.

Given $X,Y$ Hilbert modules, for each $x\in X$ and $y\in Y$, we denote $\theta_{y,x}: X\rightarrow Y$ by the map $\theta_{y,x}(z)=y\langle x,z\rangle$. This should be reminiscent of rank one operators. $\theta_{y,x}$ is adjointable with $\theta_{y,x}^*=\theta_{x,y}$. We denote $K(X,Y)=\overline{\hbox{span}}\{\theta_{y,x}: y\in Y, x\in X\}$. $K(X)$ is an ideal of $\mathcal{L}(X)$, despite in general, elements of $K(X)$ not being compact operators. To distinguish between the two notions, we say that elements of $K(X)$ are $A$-compact.

Now we say that $F\in\mathcal{L}(X,Y)$ is Fredholm, if there is a $G\in\mathcal{L}(X,Y)$ such that $1- FG$ and $1- GF$ are $A$-compact.

Atkinson-Mingo Theorem
For $F\in\mathcal{L}(X,Y)$, the following are equivalent:

1. $F$ is Fredholm,
2. There exists $K\in K(X,Y)$ such that $\tilde{F}=F+K$ has closed range and $\ker\,\tilde{F}$, $\hbox{coker}\,\tilde{F}$ are finitely generated projective $C^*$-modules over $A$.

Finally, we define $\hbox{ind}(F)=[\ker\,\tilde{F}]-[\hbox{coker}\,\tilde{F}]\in K_0(A)$.