Bounded and unbounded Fredholm operators on Hilbert modules

We briefly review some definitions and basic facts about bounded and unbounded Fredholm operators on Hilbert $C^*$-modules. We recall noncommutative version of Atiyah, Jänich and Singer theorems, and talk about path component of the space of (selfadjoint) Fredholm operators. We use representable $K$-theory and Milnor $\lim^1$- exact sequence to show that the space of Fredholm operators with coefficients in an arbitrary unital $\sigma$-$C^*$-algebra $A$, represents the functor $X\to RK_0(C(X;A))$ from the category of countably compactly generated spaces to the category of abelian groups. In particular, this shows that the Grothendieck group of $A$-vector bundles over $X$ need not be isomorphic to $[X;\mathcal F(H)]$ of homotopy classes of continuous maps from $X$ to the space of Fredholm operators on $H = l_2(A)$.