TITLE: Noncommutative Fredholm operators
QUESTION [8 upvotes]: Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely generated Hilbert $A$-modules?

REPLY [2 votes]: The image of Fredholm operator on a Hilbert module is not always closed. If it is not closed, then cokernel is certainly not finitely generated. A simple example is when $A=C_0(\mathbb{R})$ and $F$ is the diagonal matrix $\rm{diag}(x,1,1,...):H_A\to H_A$. 
However, the following two statements are true

If the image of $F$ is closed then kernel and cokernel are finitely generated.
For any Fredholm operator $F:H_A\to H_A$ there is a compact perturbation $F+K$ ($K$ is a compact operator) whose image is closed. 

See Higson, A primer on KK-theory, Theorem 3.21.