A correction to ``Adjugates in Banach algebras''

Let $A$ be a semisimple unital Banach algebra. We show that $\operatorname{rank}_A(ab)=\operatorname{rank}_A(ba)$ for all $a,b\in A$ if and only if $\operatorname{soc}(A)$ is contained in the center of $A$, and $ab\in\operatorname{soc}(A)$ implies $ba\in\operatorname{soc}(A)$ for all $a,b\in A$. This corrects an erroneous statement in R.E. Harte and C. Hernández, Adjugates in Banach algebras, Proc. Amer. Math. Soc. 134(5) (2005), 1397-1404.