Suppose $A$ is a C*-algebra. Suppose we have an approximate unit $(u_\lambda)$ then one knows that $|| u_\lambda a-au_\lambda||\rightarrow 0$ in particular $u_\lambda$ almost commutes with $a$ can one say something about the relation of $sp(u_\lambda-a)$ and $sp(u_\lambda)-sp(a)$ for large enough $\lambda$ ?
I wonder this as if $a$ and $u_\lambda$ actually commuted with each other one would have that $sp(u_\lambda-a)\subseteq sp(u_\lambda)-sp(a)$.