I've been read the book "$K$-theory and $C^*$-algebras" by Olsen. In the chapter on the generalized Fredholm index the following definitions are given:
Let $A$ be a $C^*$-algebra and $H_A$ the standard Hilbert $A$-module. Then one has defined the algebra $B(H_A)$ of bounded operators as well as the ideal $K(H_A)\subset B(H_A)$ of compact ones. Thus, one can defined the space $F_A\subset B(H_A)$ of Fredholm operators (that is, bounded operators which are invertible modulo compact operators). On the other hand, one can also define the subspace $F'_A\subset B(H_A)$ of those bounded operators $F$ such that there exists a compact perturbation $F+K$ with closed image and whose kernel and cokernel are of finite rank.
The Atkinson-Mingo theorem states that $F_A=F_A'$ if $A$ is unital. However, as I've understand, the inclusion $F_A\subset F_A'$ holds whenever $A$ is $\sigma$-unital. Is that true?
