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?
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2$\begingroup$ Could you be a bit more precise by what "Fredholm" means here? Do you mean that F is an adjointable operator on $H_A$ with finite-dimensional kernel and cokernel? Or do you have in mind a definition more "adapted" to Hilbert $C^*$-modules, such as definition on nLab? $\endgroup$– Matthew DawsCommented Mar 1, 2019 at 8:27
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3$\begingroup$ @matthew Right, I mean Fredholm in the sense of Hilbert $C^*$-module theory...that is, $F$ is an adjointable operators that is invertible modulo compact operators in $K(H_A)$. $\endgroup$– ernestCommented Mar 1, 2019 at 10:29
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$\begingroup$ +1 for your question.To be honest I was thinking to question somewhat similar to your interesting question: I submited the following note to a journal then referee inform me the shift operator is not Fredholm on $\ell^1(A)$ . Then I was thinking of consideration of the later as an $A$ module rather than a complex vector space $\endgroup$– Ali TaghaviCommented Mar 2, 2019 at 6:18
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$\begingroup$ arxiv.org/abs/1104.4196 $\endgroup$– Ali TaghaviCommented Mar 2, 2019 at 6:18
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$\begingroup$ After that the journal reject my paper with indication that the shift operator is not fredholm, I had email communication with the editor and wrote him "what about consideration $A$ module structure rather than complex vector space. $\endgroup$– Ali TaghaviCommented Mar 2, 2019 at 8:43
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1 Answer
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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.
