# Noncommutative Fredholm operators

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?

• 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? – Matthew Daws Mar 1 at 8:27
• @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 $K(H_A)$. – ernest Mar 1 at 10:29
• +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 – Ali Taghavi Mar 2 at 6:18
• arxiv.org/abs/1104.4196 – Ali Taghavi Mar 2 at 6:18
• 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. – Ali Taghavi Mar 2 at 8:43