On the continuity of the group inverse in $C^*$-algebras

Abstract

Let $\{a_n\}_{n=1}^\infty$ be a sequence of group invertible elements of a unital $C^*$ algebra $\mathcal{A}$ that converges to $a$. We present some equivalent conditions for the group invertibility of $a$ and for the convergence of $\{a_n^\#\}_{n=1}^\infty$ to $a^\#$.