limit of invertible elements in $C^*$-algebra

Question

Suppose $A$ is a non-unital $C^*$-algebra,$x$ is a positive element in $A$,what is the limit of $(x+\epsilon 1_{\tilde{A}})^{-1}$ when $\epsilon \to 0$, I guess it is $x^{-1}$,but $x$ may not be invertible.

Answer 1

This is just a sketch.

The way you formulated your questions, implies that the interval $(0,\varepsilon_0)$ is in the resultant set of $x$ for some $\varepsilon_0>0$. . If $x$ is invertible, the limit is $x^{-1}$ by continuity of the map $\varepsilon\mapsto (x+\varepsilon I)^{-1}$ in the resultant of $x$. If $x$ is not invertible, the limit does not exists, and $\|(x+\varepsilon I)^{-1}\|\xrightarrow{\varepsilon\rightarrow 0}\infty$. (See Lemma 10.17 in Rudin's functional analysis book to see what happens in the boundary of invertible elements in a Banach algebra.)